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Madrid Casado</a></div></div><div class="suggested-user-card"><div class="suggested-user-card__avatar social-profile-avatar-container"><a data-nosnippet="" href="https://cas-cz.academia.edu/BohumilVykyp%C4%9Bl"><img class="profile-avatar u-positionAbsolute" alt="Bohumil Vykypěl related author profile picture" border="0" onerror="if (this.src != '//a.academia-assets.com/images/s200_no_pic.png') this.src = '//a.academia-assets.com/images/s200_no_pic.png';" width="200" height="200" src="https://0.academia-photos.com/32096954/9588992/10681405/s200_bohumil.vykyp_l.jpeg" /></a></div><div class="suggested-user-card__user-info"><a class="suggested-user-card__user-info__header ds2-5-body-sm-bold ds2-5-body-link" href="https://cas-cz.academia.edu/BohumilVykyp%C4%9Bl">Bohumil Vykypěl</a><p class="suggested-user-card__user-info__subheader ds2-5-body-xs">Academy of Sciences of the Czech Republic</p></div></div></ul></div><style type="text/css">.suggested-academics--header h3{font-size:16px;font-weight:500;line-height:20px}</style><div class="ri-section"><div class="ri-section-header"><span>Interests</span></div><div class="ri-tags-container"></div></div></div></div><div class="right-panel-container"><div class="user-content-wrapper"><div class="uploads-container" id="social-redesign-work-container"><div class="upload-header"><h2 class="ds2-5-heading-sans-serif-xs">Uploads</h2></div><div class="documents-container backbone-social-profile-documents" style="width: 100%;"><div class="u-taCenter"></div><div class="profile--tab_content_container js-tab-pane tab-pane active" id="all"><div class="profile--tab_heading_container js-section-heading" data-section="Papers" id="Papers"><h3 class="profile--tab_heading_container">Papers by Giorgi Tutberidze</h3></div><div class="js-work-strip profile--work_container" data-work-id="113754438"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" rel="nofollow" href="https://www.academia.edu/113754438/Maximal_operators_of_T_means_with_respect_to_the_Vilenkin_system"><img alt="Research paper thumbnail of Maximal operators of T means with respect to the Vilenkin system" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title">Maximal operators of T means with respect to the Vilenkin system</div><div class="wp-workCard_item"><span>Nonlinear Studies</span><span>, Nov 24, 2020</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="113754438"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="113754438"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 113754438; 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Especially a number of wonderful results have been proved and new directions of such research has been developed e.g. concerning Wavelets Theory, Gabor Theory, Time-Frequency Analysis, Fast Fourier Transform, Abstract Harmonic Analysis, etc. One important reason for this is that this development is not only important for improving the &quot;State of the art&quot;, but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recognition). The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. 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The development of the theory of Vilenkin-Fourier series has been strongly...","internal_url":"https://www.academia.edu/93141645/A_study_of_bounded_operators_on_martingale_Hardy_spaces","translated_internal_url":"","created_at":"2022-12-18T00:56:51.048-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":127157369,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":95961550,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/95961550/thumbnails/1.jpg","file_name":"489822946.pdf","download_url":"https://www.academia.edu/attachments/95961550/download_file","bulk_download_file_name":"A_study_of_bounded_operators_on_martinga.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/95961550/489822946-libre.pdf?1671354456=\u0026response-content-disposition=attachment%3B+filename%3DA_study_of_bounded_operators_on_martinga.pdf\u0026Expires=1743686111\u0026Signature=Re5CXGcAA2N6r2K3aFSvaBYGq8bPNoxbxU9vHoDY628k1A1R0Ddu0L02uNLBohsdrggtqfVkXOjVtwUrMkerdROigiR5uQiies2F4jHPNFOpw~Dzrt6zP9STnWbIVaz4hMSEQs~JFptFnDruhB77O7POj6sWJpNRqZW8olc4H~w66gAlhOcZoO~s9K7BFpJa8PGJSE8XQNij24H7y9AaARQqtWkcoKWeEoROVrL-SE9RAtPRXtQiI1qOvHeQg1nCRizKCksHbkyOmmMy7eX8vu7j9pCqrlft372UmLryMxBssYOaVpXVd~euyg2ZlK2ay58LdDX97ja87FxTw0oYVQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_study_of_bounded_operators_on_martingale_Hardy_spaces","translated_slug":"","page_count":171,"language":"en","content_type":"Work","summary":"The classical Fourier Analysis has been developed in an almost unbelievable way from the first fundamental discoveries by Fourier. Especially a number of wonderful results have been proved and new directions of such research has been developed e.g. concerning Wavelets Theory, Gabor Theory, Time-Frequency Analysis, Fast Fourier Transform, Abstract Harmonic Analysis, etc. One important reason for this is that this development is not only important for improving the \u0026quot;State of the art\u0026quot;, but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recognition). The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly...","owner":{"id":127157369,"first_name":"Giorgi","middle_initials":null,"last_name":"Tutberidze","page_name":"Tutberidze","domain_name":"independent","created_at":"2019-09-19T03:49:23.296-07:00","display_name":"Giorgi Tutberidze","url":"https://independent.academia.edu/Tutberidze"},"attachments":[{"id":95961550,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/95961550/thumbnails/1.jpg","file_name":"489822946.pdf","download_url":"https://www.academia.edu/attachments/95961550/download_file","bulk_download_file_name":"A_study_of_bounded_operators_on_martinga.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/95961550/489822946-libre.pdf?1671354456=\u0026response-content-disposition=attachment%3B+filename%3DA_study_of_bounded_operators_on_martinga.pdf\u0026Expires=1743686111\u0026Signature=Re5CXGcAA2N6r2K3aFSvaBYGq8bPNoxbxU9vHoDY628k1A1R0Ddu0L02uNLBohsdrggtqfVkXOjVtwUrMkerdROigiR5uQiies2F4jHPNFOpw~Dzrt6zP9STnWbIVaz4hMSEQs~JFptFnDruhB77O7POj6sWJpNRqZW8olc4H~w66gAlhOcZoO~s9K7BFpJa8PGJSE8XQNij24H7y9AaARQqtWkcoKWeEoROVrL-SE9RAtPRXtQiI1qOvHeQg1nCRizKCksHbkyOmmMy7eX8vu7j9pCqrlft372UmLryMxBssYOaVpXVd~euyg2ZlK2ay58LdDX97ja87FxTw0oYVQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"},{"id":95961551,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/95961551/thumbnails/1.jpg","file_name":"489822946.pdf","download_url":"https://www.academia.edu/attachments/95961551/download_file","bulk_download_file_name":"A_study_of_bounded_operators_on_martinga.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/95961551/489822946-libre.pdf?1671354470=\u0026response-content-disposition=attachment%3B+filename%3DA_study_of_bounded_operators_on_martinga.pdf\u0026Expires=1743686111\u0026Signature=BBYAm31NoymKarrOVRhqDmYb0Hd7kspa3axZiYivKa1CYPYV511-pvduVWyhYH36HhvoT~Khk04xCp9GRfSPZ5xgrlSbx892zZYNKTVpOaXyXZIsLRzvxVA4xnRvkgUKwFbXiuM1lOgA74q-jYrr6M4Kfz4d7buFgJXYnYQG7k1JAzhieAGz2u8WTxQ9JlU--wnVlQHG8bsq5m3eIYZQlhDf7lo~6etBf0mt1BjnNYyFzadI6Ccq1IvFe8BqzP779gLn02KAkv6nrYQfR48i0DjfkhildJyY7NLmO~huj4Qqtx6n2kaLjQVafqJeTWMlBSXvq40wgyeT8~0MW-aBCA__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":1188947,"name":"D","url":"https://www.academia.edu/Documents/in/D-351414216"},{"id":1188997,"name":"C","url":"https://www.academia.edu/Documents/in/C"},{"id":1194168,"name":"B","url":"https://www.academia.edu/Documents/in/B"},{"id":2760784,"name":"doktor","url":"https://www.academia.edu/Documents/in/doktor"}],"urls":[{"id":27154931,"url":"https://core.ac.uk/download/489822946.pdf"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") if (false) { Aedu.setUpFigureCarousel('profile-work-93141645-figures'); } }); </script> <div class="js-work-strip profile--work_container" data-work-id="86377259"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/86377259/Bounded_operators_on_Martingale_Hardy_spaces"><img alt="Research paper thumbnail of Bounded operators on Martingale Hardy spaces" class="work-thumbnail" src="https://attachments.academia-assets.com/90843464/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377259/Bounded_operators_on_Martingale_Hardy_spaces">Bounded operators on Martingale Hardy spaces</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. In addition, we investigate some $T$ means, which are &quot;inverse&quot; summability methods of Nörlund, but only in the case when their coefficients are monotone.</span></div><div class="wp-workCard_item"><div class="carousel-container carousel-container--sm" id="profile-work-86377259-figures"><div class="prev-slide-container js-prev-button-container"><button aria-label="Previous" class="carousel-navigation-button js-profile-work-86377259-figures-prev"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">arrow_back_ios</span></button></div><div class="slides-container js-slides-container"><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431783/figure-20-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_020.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431772/figure-17-proof-since-sequence-is-non-increasing-number-we"><img alt="Proof: Since sequence is non-increasing number we get that If we apply (3.15) and (3.16) in Lemma 3.1 we immediately get that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_017.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431777/figure-18-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_018.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431779/figure-19-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_019.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431786/figure-21-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_021.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431787/figure-22-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_022.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431789/figure-23-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_023.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431790/figure-24-without-lost-the-generality-we-may-assume-that-in"><img alt="Without lost the generality we may assume that c = 1 in (3.23). By combining (3.32) and (3.33) we get " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_024.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431791/figure-25-number-of-special-cases-of-our-results-are-of"><img alt="A number of special cases of our results are of particular interest and give both well- known and new information. We just give the following examples of such 7’ means with non-increasing sequence {q, : k > 0}: " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_025.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431763/figure-13-lemma-let-and-be-sequence-of-non-increasing"><img alt="Lemma 3.2 Let n © N and {q, : k © N} be a sequence of non-increasing numbers, or non-decreasing function satisfying condition (3.13). Then Moreover, if f € Hy/2 and {a : k > 0} be a sequence of non-increasing numbers, sat- isfying the condition (3.13), then there exists an absolute constant c, such that the inequality " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_013.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431765/figure-14-lemma-let-be-sequence-of-non-increasing-numbers"><img alt="Lemma 3.3 Let {q, : k € N} be a sequence of non-increasing numbers andn > My. Ther " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_014.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431768/figure-15-proof-since-sequence-is-non-increasing-number-we"><img alt="Proof: Since sequence is non-increasing number we get that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_015.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431739/figure-1-it-is-well-known-that-for-all"><img alt="It is well-known that for all n € N, " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_001.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431741/figure-2-we-state-well-known-equalities-for-dirichlet"><img alt="We state well-known equalities for Dirichlet kernels (for details see e.g. [62] and [112]): is finite. It is well known that L” is a Banach space for each 1 < p < on. is finite. Let L© represent the collection of f € L° for which " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_002.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431743/figure-3-kernels-of-norlund-logarithmic-mean-and-riesz"><img alt="Kernels of Norlund logarithmic mean and Riesz logarithmic mean are respectively de- ~~. J ke, " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_003.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431744/figure-4-as-corollary-we-also-get-that-that-if-hy-then"><img alt="As a corollary we also get that that if f € Hy /2, then " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_004.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431745/figure-5-in-blahota-and-tephnadze-also-considered-the"><img alt="In Blahota and Tephnadze [13] also considered the endpoint case p = 1/2 and they proved that if f € H,2 then there exists an absolute constant c such that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_005.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431746/figure-6-since-if-we-use-the-estimates-above-then-we-obtain"><img alt="Since m;_; <1; — 2 if we use the estimates above, then we obtain that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_006.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431747/figure-7-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_007.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431749/figure-8-hence-by-applying-we-find-that"><img alt="Hence, by applying (1.14) we find that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_008.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431751/figure-9-without-loss-the-generality-we-may-assume-that-let"><img alt="Without loss the generality we may assume that? < 7. Let x € ie and 7 < (nz). Then x —t € I,? fort € Iy and, according to (1.9), we obtain that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_009.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431755/figure-10-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_010.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431756/figure-11-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_011.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431760/figure-12-hence-for-sufficiently-large-we-can-write-that"><img alt="Hence, for sufficiently large k, we can write that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_012.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431769/figure-16-lemma-let-my-and-be-sequence-of-non-increasing"><img alt="Lemma 3.5 Let n > My and {q, : k © N} be a sequence of non-increasing number satisfying condition (3.43). Then " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_016.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431793/figure-26-if-we-apply-according-that-is-bounded-from-the"><img alt="If we apply (3.38), according that o* is bounded from the Hardy space Hj 2 to the space weak — L1/2, we can conclude that the maximal operators 7™* of all J’ means with non- decreasing sequence {q, : k > 0} satisfying the condition 3.37 are bounded from the Hardy space Hj /2 to the space weak — L1/2. The proof of part a) is complete. According to (3.34), (3.35) and (3.36) we can conclude that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_026.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431794/figure-27-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_027.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431796/figure-28-proof-let-the-sequence-be-non-decreasing"><img alt="Proof: Let the sequence {q, : k > 0} be non-decreasing satisfying the condition (3.40). 3y combining (3.15) and (3.17) we get that Then all such T means are bounded from the Hardy space H,, to the space L, " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_028.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431798/figure-29-let-by-using-we-find-that"><img alt="Let 0 < p < 1/2. By using (1.1), (3.47), (3.48) we find that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_029.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431800/figure-30-let-and-be-means-with-non-increasing-coefficients"><img alt="Let p = 1/2 and T;, be T means, with non-increasing coefficients {q, : k > O}, sat- isfying condition (3.43). By Lemma 1.3, the proof of part b) will be complete, if we show that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_030.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431802/figure-31-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_031.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431804/figure-32-hence-by-simple-calculation-we-can-conclude-that"><img alt="Hence, by simple calculation we can conclude that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_032.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431806/figure-33-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_033.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431807/figure-34-by-combining-for-and-we-find-that"><img alt="By combining (4.2), (4.23)-(4.28) for € I2(e9 + e1) and 0 < p < 1/2 we find that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_034.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431809/figure-35-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_035.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431811/figure-36-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_036.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431813/figure-37-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_037.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431817/figure-38-for-all-lebesgue-points-of-by-using-we-can"><img alt="for all Lebesgue points of f € L,(G,,). By using (1.2) we can conclude that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_038.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431822/table-1-the-proof-of-lemma-can-be-found-in-tephnadze"><img alt="The proof of Lemma can be found in Tephnadze [143]. 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In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. In addition, we investigate some $T$ means, which are \u0026quot;inverse\u0026quot; summability methods of Nörlund, but only in the case when their coefficients are monotone.","publisher":"arXiv","ai_title_tag":"Bounded Operators and Convergence in Martingale Hardy Spaces","publication_date":{"day":null,"month":null,"year":2022,"errors":{}}},"translated_abstract":"The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. 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In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. In addition, we investigate some $T$ means, which are \u0026quot;inverse\u0026quot; summability methods of Nörlund, but only in the case when their coefficients are monotone.","owner":{"id":127157369,"first_name":"Giorgi","middle_initials":null,"last_name":"Tutberidze","page_name":"Tutberidze","domain_name":"independent","created_at":"2019-09-19T03:49:23.296-07:00","display_name":"Giorgi Tutberidze","url":"https://independent.academia.edu/Tutberidze"},"attachments":[{"id":90843464,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/90843464/thumbnails/1.jpg","file_name":"2201.12134v3.pdf","download_url":"https://www.academia.edu/attachments/90843464/download_file","bulk_download_file_name":"Bounded_operators_on_Martingale_Hardy_sp.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/90843464/2201.12134v3-libre.pdf?1662763582=\u0026response-content-disposition=attachment%3B+filename%3DBounded_operators_on_Martingale_Hardy_sp.pdf\u0026Expires=1743686111\u0026Signature=Iy~7cRdk0PL~f0vGbC6q-9a2PzCo-syOdI3Dlp5SbOGi6gtIue-~AN8pjQy~1yZ4IBRPX-PDG-ONHYgZGKfXCI7FUrFNLCdYDbaI8pYFnGEs5v7LRbp41zt8TCz7vBLQjHXfaVcjkKwVrEbOXE11nrOYFEvC3RLYyvo0eVq27wi~z0J7JAtY6LQ~nOfej5UTlLQ9J4g00Fl7CABaftNZAqmfg7uQxJHXzPrFCHIgwrdEvybWqEGieDue~OeGaeGVdwSR3Svb5rF3gaiuaBw59hSOxr3MfRCt5OG-8J~ltk2-Z9k6~8jIbDkUfp3A6HUvMwXWJI~G1QyjtFLqhdWkcg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":893194,"name":"Inverse","url":"https://www.academia.edu/Documents/in/Inverse"},{"id":1228829,"name":"Fourier Series","url":"https://www.academia.edu/Documents/in/Fourier_Series"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") if (true) { Aedu.setUpFigureCarousel('profile-work-86377259-figures'); } }); </script> <div class="js-work-strip profile--work_container" data-work-id="86377258"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/86377258/a_e_Convergence_of_N%C3%B6rlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions"><img alt="Research paper thumbnail of a.e. Convergence of Nörlund means with respect to Vilenkin systems of integrable functions" class="work-thumbnail" src="https://attachments.academia-assets.com/90843430/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377258/a_e_Convergence_of_N%C3%B6rlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions">a.e. Convergence of Nörlund means with respect to Vilenkin systems of integrable functions</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coeffi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise and norm convergence in L_p norms of such Nörlund means.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="adef441297e20248975f6c10a132c006" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":90843430,"asset_id":86377258,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/90843430/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="86377258"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="86377258"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 86377258; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=86377258]").text(description); $(".js-view-count[data-work-id=86377258]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 86377258; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='86377258']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "adef441297e20248975f6c10a132c006" } } $('.js-work-strip[data-work-id=86377258]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":86377258,"title":"a.e. Convergence of Nörlund means with respect to Vilenkin systems of integrable functions","translated_title":"","metadata":{"abstract":"In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise and norm convergence in L_p norms of such Nörlund means.","publication_date":{"day":1,"month":7,"year":2021,"errors":{}}},"translated_abstract":"In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise and norm convergence in L_p norms of such Nörlund means.","internal_url":"https://www.academia.edu/86377258/a_e_Convergence_of_N%C3%B6rlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions","translated_internal_url":"","created_at":"2022-09-09T15:12:32.410-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":127157369,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":90843430,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/90843430/thumbnails/1.jpg","file_name":"2107.02898v1.pdf","download_url":"https://www.academia.edu/attachments/90843430/download_file","bulk_download_file_name":"a_e_Convergence_of_Norlund_means_with_re.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/90843430/2107.02898v1-libre.pdf?1662763563=\u0026response-content-disposition=attachment%3B+filename%3Da_e_Convergence_of_Norlund_means_with_re.pdf\u0026Expires=1743686111\u0026Signature=DAYa-MXAoNoPpqHTFNsn6haNeDxGb9-fudRpX5L~zzSIhKIfv2cea6hVEzF6slchcIXw8wWLQ-yqb9XA12Jd~HXTER6G5e5L-nDxP-ZL5SiZXG65zU9wVEXdlnACzpM2jd-z2C-uGdb7LPN3mPobREo3~AI65A1FfgIS5BRUzSi1~D3Odmk7nK0LrE9an2auT~stms3pd4wqhFhlpAAdlOgrRQZ8P~fIBvt1ZiYT2OtZyC2oF0-qTrhHbpIfLb5JIyrdfsVGlpMlJveY6FN9ZXx-l9GZ2noQTsw7Wo~1m3s04Wd5OWWw8P8N-jONX5v4Fkr4I6BQqHSs20KGnB-xSw__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"a_e_Convergence_of_Nörlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions","translated_slug":"","page_count":13,"language":"en","content_type":"Work","summary":"In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") if (false) { Aedu.setUpFigureCarousel('profile-work-86377258-figures'); } }); </script> <div class="js-work-strip profile--work_container" data-work-id="86377257"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/86377257/On_the_strong_convergence_of_partial_sums_with_respect_to_bounded_Vilenkin_systems"><img alt="Research paper thumbnail of On the strong convergence of partial sums with respect to bounded Vilenkin systems" class="work-thumbnail" src="https://attachments.academia-assets.com/90843462/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377257/On_the_strong_convergence_of_partial_sums_with_respect_to_bounded_Vilenkin_systems">On the strong convergence of partial sums with respect to bounded Vilenkin systems</a></div><div class="wp-workCard_item"><span>arXiv: Classical Analysis and ODEs</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we investigate some strong convergence theorems for partial sums with respect to Vi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6d76064d19bae819feb00b12fc6e2e55" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":90843462,"asset_id":86377257,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/90843462/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="86377257"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="86377257"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 86377257; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") if (false) { Aedu.setUpFigureCarousel('profile-work-86377257-figures'); } }); </script> <div class="js-work-strip profile--work_container" data-work-id="86377256"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/86377256/Lebesgue_and_Vilenkin_Lebesgue_points_and_a_e_Convergence_of_N_orlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions"><img alt="Research paper thumbnail of Lebesgue and Vilenkin-Lebesgue points and a. e. Convergence of N\"orlund means with respect to Vilenkin systems of integrable functions" class="work-thumbnail" src="https://attachments.academia-assets.com/90843431/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377256/Lebesgue_and_Vilenkin_Lebesgue_points_and_a_e_Convergence_of_N_orlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions">Lebesgue and Vilenkin-Lebesgue points and a. e. Convergence of N\"orlund means with respect to Vilenkin systems of integrable functions</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coeffi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise convergence and convergence in Lp norms of such Nörlund means. 2000 Mathematics Subject Classification. 42C10, 42B25.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="10a81e4bf46504697ce844564f904220" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":90843431,"asset_id":86377256,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/90843431/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="86377256"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="86377256"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 86377256; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=86377256]").text(description); $(".js-view-count[data-work-id=86377256]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 86377256; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='86377256']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "10a81e4bf46504697ce844564f904220" } } $('.js-work-strip[data-work-id=86377256]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":86377256,"title":"Lebesgue and Vilenkin-Lebesgue points and a. e. 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Moreover, we discuss pointwise convergence and convergence in Lp norms of such Nörlund means. 2000 Mathematics Subject Classification. 42C10, 42B25.","internal_url":"https://www.academia.edu/86377256/Lebesgue_and_Vilenkin_Lebesgue_points_and_a_e_Convergence_of_N_orlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions","translated_internal_url":"","created_at":"2022-09-09T15:12:32.015-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":127157369,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":90843431,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/90843431/thumbnails/1.jpg","file_name":"2107.02898v2.pdf","download_url":"https://www.academia.edu/attachments/90843431/download_file","bulk_download_file_name":"Lebesgue_and_Vilenkin_Lebesgue_points_an.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/90843431/2107.02898v2-libre.pdf?1662763567=\u0026response-content-disposition=attachment%3B+filename%3DLebesgue_and_Vilenkin_Lebesgue_points_an.pdf\u0026Expires=1743686111\u0026Signature=bDHxO~zTvOVlt1zDUSQJMtPvWfcwiRMArXpRrME2YwLUfNZGzkpY4GN23~YRn5TMSKYv2thDg6txtPQKnZtDD-cRUa59Fs1ejvozO6EKWPwo-OeefKkqkZr5J0UEz9C8l916EtkE07HAYTnrShTxpspyy-ku~x9cmaWWvwWNOuM04yqtAAD44m7gYJbKHkqD~5fZmaN9a2GtPUhjGKGe98uPGqx4QSzYL2pEsl4853Hu17vDJQWh8S1etRQu6OVsWZoOLozYXjApzU4vZ3uuv1dkzUzLfpc8SFw0SO-WPzxoL-ILrW0qUGG37~An8kKiHsIkRCtp-5NPnHj1Z0VwWg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Lebesgue_and_Vilenkin_Lebesgue_points_and_a_e_Convergence_of_N_orlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions","translated_slug":"","page_count":13,"language":"en","content_type":"Work","summary":"In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise convergence and convergence in Lp norms of such Nörlund means. 2000 Mathematics Subject Classification. 42C10, 42B25.","owner":{"id":127157369,"first_name":"Giorgi","middle_initials":null,"last_name":"Tutberidze","page_name":"Tutberidze","domain_name":"independent","created_at":"2019-09-19T03:49:23.296-07:00","display_name":"Giorgi 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data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377210/A_note_on_The_maximal_operators_of_the_N_orlund_logaritmic_means_of_Vilenkin_Fourier_series">A note on The maximal operators of the N\"orlund logaritmic means of Vilenkin-Fourier series</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The main aim of this paper is to investigate $\left(H_{p},L_{p}\right)$- type inequalities for th...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The main aim of this paper is to investigate $\left(H_{p},L_{p}\right)$- type inequalities for the the maximal operators of N\&quot;orlund logaritmic means, for $0</span></div><div class="wp-workCard_item wp-workCard--actions"><span 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wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377209/Some_inequalities_related_to_strong_convergence_of_Riesz_logarithmic_means">Some inequalities related to strong convergence of Riesz logarithmic means</a></div><div class="wp-workCard_item"><span>Journal of Inequalities and Applications</span><span>, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-di...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin–Fourier (Walsh–Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh–Fourier series.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a6fb49d11fe86df3894d5687a1fcfae2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":90843386,"asset_id":86377209,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/90843386/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="86377209"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="86377209"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 86377209; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=86377209]").text(description); $(".js-view-count[data-work-id=86377209]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 86377209; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='86377209']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a6fb49d11fe86df3894d5687a1fcfae2" } } $('.js-work-strip[data-work-id=86377209]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":86377209,"title":"Some inequalities related to strong convergence of Riesz logarithmic means","translated_title":"","metadata":{"abstract":"In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin–Fourier (Walsh–Fourier) series. 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In particular, we found special conditions for the functions of orthonormal system (ONS), for which the above sequences are absolute convergence factors of Fourier series of functions of Lip ⁡ 1 {\operatorname{Lip}1} class. 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Especially a number of wonderful results have been proved and new directions of such research has been developed e.g. concerning Wavelets Theory, Gabor Theory, Time-Frequency Analysis, Fast Fourier Transform, Abstract Harmonic Analysis, etc. One important reason for this is that this development is not only important for improving the &quot;State of the art&quot;, but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recognition). The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly...</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="3ecfdf23b2d83a5e48d74129e5a9be92" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":95961550,"asset_id":93141645,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/95961550/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="93141645"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="93141645"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 93141645; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=93141645]").text(description); $(".js-view-count[data-work-id=93141645]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 93141645; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='93141645']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "3ecfdf23b2d83a5e48d74129e5a9be92" } } $('.js-work-strip[data-work-id=93141645]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":93141645,"title":"A study of bounded operators on martingale Hardy spaces","translated_title":"","metadata":{"abstract":"The classical Fourier Analysis has been developed in an almost unbelievable way from the first fundamental discoveries by Fourier. Especially a number of wonderful results have been proved and new directions of such research has been developed e.g. concerning Wavelets Theory, Gabor Theory, Time-Frequency Analysis, Fast Fourier Transform, Abstract Harmonic Analysis, etc. One important reason for this is that this development is not only important for improving the \u0026quot;State of the art\u0026quot;, but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recognition). The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly...","publisher":"'Allerton Press'","publication_date":{"day":13,"month":12,"year":2021,"errors":{}}},"translated_abstract":"The classical Fourier Analysis has been developed in an almost unbelievable way from the first fundamental discoveries by Fourier. Especially a number of wonderful results have been proved and new directions of such research has been developed e.g. concerning Wavelets Theory, Gabor Theory, Time-Frequency Analysis, Fast Fourier Transform, Abstract Harmonic Analysis, etc. One important reason for this is that this development is not only important for improving the \u0026quot;State of the art\u0026quot;, but also for its importance in other areas of mathematics and also for several applications (e.g. theory of signal transmission, multiplexing, filtering, image enhancement, coding theory, digital signal processing and pattern recognition). The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. The development of the theory of Vilenkin-Fourier series has been strongly...","internal_url":"https://www.academia.edu/93141645/A_study_of_bounded_operators_on_martingale_Hardy_spaces","translated_internal_url":"","created_at":"2022-12-18T00:56:51.048-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":127157369,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":95961550,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/95961550/thumbnails/1.jpg","file_name":"489822946.pdf","download_url":"https://www.academia.edu/attachments/95961550/download_file","bulk_download_file_name":"A_study_of_bounded_operators_on_martinga.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/95961550/489822946-libre.pdf?1671354456=\u0026response-content-disposition=attachment%3B+filename%3DA_study_of_bounded_operators_on_martinga.pdf\u0026Expires=1743686111\u0026Signature=Re5CXGcAA2N6r2K3aFSvaBYGq8bPNoxbxU9vHoDY628k1A1R0Ddu0L02uNLBohsdrggtqfVkXOjVtwUrMkerdROigiR5uQiies2F4jHPNFOpw~Dzrt6zP9STnWbIVaz4hMSEQs~JFptFnDruhB77O7POj6sWJpNRqZW8olc4H~w66gAlhOcZoO~s9K7BFpJa8PGJSE8XQNij24H7y9AaARQqtWkcoKWeEoROVrL-SE9RAtPRXtQiI1qOvHeQg1nCRizKCksHbkyOmmMy7eX8vu7j9pCqrlft372UmLryMxBssYOaVpXVd~euyg2ZlK2ay58LdDX97ja87FxTw0oYVQ__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"A_study_of_bounded_operators_on_martingale_Hardy_spaces","translated_slug":"","page_count":171,"language":"en","content_type":"Work","summary":"The classical Fourier Analysis has been developed in an almost unbelievable way from the first fundamental discoveries by Fourier. 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In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. In addition, we investigate some $T$ means, which are &quot;inverse&quot; summability methods of Nörlund, but only in the case when their coefficients are monotone.</span></div><div class="wp-workCard_item"><div class="carousel-container carousel-container--sm" id="profile-work-86377259-figures"><div class="prev-slide-container js-prev-button-container"><button aria-label="Previous" class="carousel-navigation-button js-profile-work-86377259-figures-prev"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">arrow_back_ios</span></button></div><div class="slides-container js-slides-container"><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431783/figure-20-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_020.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431772/figure-17-proof-since-sequence-is-non-increasing-number-we"><img alt="Proof: Since sequence is non-increasing number we get that If we apply (3.15) and (3.16) in Lemma 3.1 we immediately get that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_017.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431777/figure-18-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_018.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431779/figure-19-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_019.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431786/figure-21-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_021.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431787/figure-22-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_022.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431789/figure-23-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_023.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431790/figure-24-without-lost-the-generality-we-may-assume-that-in"><img alt="Without lost the generality we may assume that c = 1 in (3.23). By combining (3.32) and (3.33) we get " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_024.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431791/figure-25-number-of-special-cases-of-our-results-are-of"><img alt="A number of special cases of our results are of particular interest and give both well- known and new information. We just give the following examples of such 7’ means with non-increasing sequence {q, : k > 0}: " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_025.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431763/figure-13-lemma-let-and-be-sequence-of-non-increasing"><img alt="Lemma 3.2 Let n © N and {q, : k © N} be a sequence of non-increasing numbers, or non-decreasing function satisfying condition (3.13). Then Moreover, if f € Hy/2 and {a : k > 0} be a sequence of non-increasing numbers, sat- isfying the condition (3.13), then there exists an absolute constant c, such that the inequality " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_013.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431765/figure-14-lemma-let-be-sequence-of-non-increasing-numbers"><img alt="Lemma 3.3 Let {q, : k € N} be a sequence of non-increasing numbers andn > My. Ther " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_014.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431768/figure-15-proof-since-sequence-is-non-increasing-number-we"><img alt="Proof: Since sequence is non-increasing number we get that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_015.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431739/figure-1-it-is-well-known-that-for-all"><img alt="It is well-known that for all n € N, " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_001.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431741/figure-2-we-state-well-known-equalities-for-dirichlet"><img alt="We state well-known equalities for Dirichlet kernels (for details see e.g. [62] and [112]): is finite. It is well known that L” is a Banach space for each 1 < p < on. is finite. Let L© represent the collection of f € L° for which " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_002.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431743/figure-3-kernels-of-norlund-logarithmic-mean-and-riesz"><img alt="Kernels of Norlund logarithmic mean and Riesz logarithmic mean are respectively de- ~~. J ke, " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_003.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431744/figure-4-as-corollary-we-also-get-that-that-if-hy-then"><img alt="As a corollary we also get that that if f € Hy /2, then " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_004.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431745/figure-5-in-blahota-and-tephnadze-also-considered-the"><img alt="In Blahota and Tephnadze [13] also considered the endpoint case p = 1/2 and they proved that if f € H,2 then there exists an absolute constant c such that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_005.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431746/figure-6-since-if-we-use-the-estimates-above-then-we-obtain"><img alt="Since m;_; <1; — 2 if we use the estimates above, then we obtain that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_006.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431747/figure-7-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_007.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431749/figure-8-hence-by-applying-we-find-that"><img alt="Hence, by applying (1.14) we find that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_008.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431751/figure-9-without-loss-the-generality-we-may-assume-that-let"><img alt="Without loss the generality we may assume that? < 7. Let x € ie and 7 < (nz). Then x —t € I,? fort € Iy and, according to (1.9), we obtain that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_009.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431755/figure-10-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_010.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431756/figure-11-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_011.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431760/figure-12-hence-for-sufficiently-large-we-can-write-that"><img alt="Hence, for sufficiently large k, we can write that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_012.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431769/figure-16-lemma-let-my-and-be-sequence-of-non-increasing"><img alt="Lemma 3.5 Let n > My and {q, : k © N} be a sequence of non-increasing number satisfying condition (3.43). Then " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_016.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431793/figure-26-if-we-apply-according-that-is-bounded-from-the"><img alt="If we apply (3.38), according that o* is bounded from the Hardy space Hj 2 to the space weak — L1/2, we can conclude that the maximal operators 7™* of all J’ means with non- decreasing sequence {q, : k > 0} satisfying the condition 3.37 are bounded from the Hardy space Hj /2 to the space weak — L1/2. The proof of part a) is complete. According to (3.34), (3.35) and (3.36) we can conclude that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_026.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431794/figure-27-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_027.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431796/figure-28-proof-let-the-sequence-be-non-decreasing"><img alt="Proof: Let the sequence {q, : k > 0} be non-decreasing satisfying the condition (3.40). 3y combining (3.15) and (3.17) we get that Then all such T means are bounded from the Hardy space H,, to the space L, " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_028.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431798/figure-29-let-by-using-we-find-that"><img alt="Let 0 < p < 1/2. By using (1.1), (3.47), (3.48) we find that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_029.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431800/figure-30-let-and-be-means-with-non-increasing-coefficients"><img alt="Let p = 1/2 and T;, be T means, with non-increasing coefficients {q, : k > O}, sat- isfying condition (3.43). By Lemma 1.3, the proof of part b) will be complete, if we show that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_030.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431802/figure-31-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_031.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431804/figure-32-hence-by-simple-calculation-we-can-conclude-that"><img alt="Hence, by simple calculation we can conclude that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_032.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431806/figure-33-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_033.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431807/figure-34-by-combining-for-and-we-find-that"><img alt="By combining (4.2), (4.23)-(4.28) for € I2(e9 + e1) and 0 < p < 1/2 we find that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_034.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431809/figure-35-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_035.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431811/figure-36-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_036.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431813/figure-37-bounded-operators-on-martingale-hardy-spaces"><img alt="" class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_037.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431817/figure-38-for-all-lebesgue-points-of-by-using-we-can"><img alt="for all Lebesgue points of f € L,(G,,). By using (1.2) we can conclude that " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/figure_038.jpg" /></a></figure><figure class="figure-slide-container"><a href="https://www.academia.edu/figures/33431822/table-1-the-proof-of-lemma-can-be-found-in-tephnadze"><img alt="The proof of Lemma can be found in Tephnadze [143]. " class="figure-slide-image" src="https://figures.academia-assets.com/90843464/table_001.jpg" /></a></figure></div><div class="next-slide-container js-next-button-container"><button aria-label="Next" class="carousel-navigation-button js-profile-work-86377259-figures-next"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">arrow_forward_ios</span></button></div></div></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="9fff3e7b8016f2b7e0251fcebf4ab21f" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":90843464,"asset_id":86377259,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/90843464/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="86377259"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="86377259"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 86377259; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=86377259]").text(description); $(".js-view-count[data-work-id=86377259]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 86377259; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='86377259']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "9fff3e7b8016f2b7e0251fcebf4ab21f" } } $('.js-work-strip[data-work-id=86377259]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":86377259,"title":"Bounded operators on Martingale Hardy spaces","translated_title":"","metadata":{"abstract":"The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. In addition, we investigate some $T$ means, which are \u0026quot;inverse\u0026quot; summability methods of Nörlund, but only in the case when their coefficients are monotone.","publisher":"arXiv","ai_title_tag":"Bounded Operators and Convergence in Martingale Hardy Spaces","publication_date":{"day":null,"month":null,"year":2022,"errors":{}}},"translated_abstract":"The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. In addition, we investigate some $T$ means, which are \u0026quot;inverse\u0026quot; summability methods of Nörlund, but only in the case when their coefficients are monotone.","internal_url":"https://www.academia.edu/86377259/Bounded_operators_on_Martingale_Hardy_spaces","translated_internal_url":"","created_at":"2022-09-09T15:12:32.604-07:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":127157369,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[{"id":90843464,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/90843464/thumbnails/1.jpg","file_name":"2201.12134v3.pdf","download_url":"https://www.academia.edu/attachments/90843464/download_file","bulk_download_file_name":"Bounded_operators_on_Martingale_Hardy_sp.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/90843464/2201.12134v3-libre.pdf?1662763582=\u0026response-content-disposition=attachment%3B+filename%3DBounded_operators_on_Martingale_Hardy_sp.pdf\u0026Expires=1743686111\u0026Signature=Iy~7cRdk0PL~f0vGbC6q-9a2PzCo-syOdI3Dlp5SbOGi6gtIue-~AN8pjQy~1yZ4IBRPX-PDG-ONHYgZGKfXCI7FUrFNLCdYDbaI8pYFnGEs5v7LRbp41zt8TCz7vBLQjHXfaVcjkKwVrEbOXE11nrOYFEvC3RLYyvo0eVq27wi~z0J7JAtY6LQ~nOfej5UTlLQ9J4g00Fl7CABaftNZAqmfg7uQxJHXzPrFCHIgwrdEvybWqEGieDue~OeGaeGVdwSR3Svb5rF3gaiuaBw59hSOxr3MfRCt5OG-8J~ltk2-Z9k6~8jIbDkUfp3A6HUvMwXWJI~G1QyjtFLqhdWkcg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"slug":"Bounded_operators_on_Martingale_Hardy_spaces","translated_slug":"","page_count":134,"language":"en","content_type":"Work","summary":"The aim of my thesis is to discuss, develop and apply the newest developments of this fascinating theory connected to modern harmonic analysis. In particular, we investigate some strong convergence result of partial sums of Vilenkin-Fourier series. Moreover, we derive necessary and sufficient conditions for the modulus of continuity so that norm convergence of subsequences of Fejér means is valid. Furthermore, we consider Riesz and Nörlund logarithmic means. It is also proved that these results are the best possible in a special sense. As applications both some well-known and new results are pointed out. In addition, we investigate some $T$ means, which are \u0026quot;inverse\u0026quot; summability methods of Nörlund, but only in the case when their coefficients are monotone.","owner":{"id":127157369,"first_name":"Giorgi","middle_initials":null,"last_name":"Tutberidze","page_name":"Tutberidze","domain_name":"independent","created_at":"2019-09-19T03:49:23.296-07:00","display_name":"Giorgi Tutberidze","url":"https://independent.academia.edu/Tutberidze"},"attachments":[{"id":90843464,"title":"","file_type":"pdf","scribd_thumbnail_url":"https://attachments.academia-assets.com/90843464/thumbnails/1.jpg","file_name":"2201.12134v3.pdf","download_url":"https://www.academia.edu/attachments/90843464/download_file","bulk_download_file_name":"Bounded_operators_on_Martingale_Hardy_sp.pdf","bulk_download_url":"https://d1wqtxts1xzle7.cloudfront.net/90843464/2201.12134v3-libre.pdf?1662763582=\u0026response-content-disposition=attachment%3B+filename%3DBounded_operators_on_Martingale_Hardy_sp.pdf\u0026Expires=1743686111\u0026Signature=Iy~7cRdk0PL~f0vGbC6q-9a2PzCo-syOdI3Dlp5SbOGi6gtIue-~AN8pjQy~1yZ4IBRPX-PDG-ONHYgZGKfXCI7FUrFNLCdYDbaI8pYFnGEs5v7LRbp41zt8TCz7vBLQjHXfaVcjkKwVrEbOXE11nrOYFEvC3RLYyvo0eVq27wi~z0J7JAtY6LQ~nOfej5UTlLQ9J4g00Fl7CABaftNZAqmfg7uQxJHXzPrFCHIgwrdEvybWqEGieDue~OeGaeGVdwSR3Svb5rF3gaiuaBw59hSOxr3MfRCt5OG-8J~ltk2-Z9k6~8jIbDkUfp3A6HUvMwXWJI~G1QyjtFLqhdWkcg__\u0026Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA"}],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":893194,"name":"Inverse","url":"https://www.academia.edu/Documents/in/Inverse"},{"id":1228829,"name":"Fourier Series","url":"https://www.academia.edu/Documents/in/Fourier_Series"}],"urls":[]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") if (true) { Aedu.setUpFigureCarousel('profile-work-86377259-figures'); } }); </script> <div class="js-work-strip profile--work_container" data-work-id="86377258"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/86377258/a_e_Convergence_of_N%C3%B6rlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions"><img alt="Research paper thumbnail of a.e. Convergence of Nörlund means with respect to Vilenkin systems of integrable functions" class="work-thumbnail" src="https://attachments.academia-assets.com/90843430/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377258/a_e_Convergence_of_N%C3%B6rlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions">a.e. Convergence of Nörlund means with respect to Vilenkin systems of integrable functions</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coeffi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise and norm convergence in L_p norms of such Nörlund means.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="adef441297e20248975f6c10a132c006" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":90843430,"asset_id":86377258,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/90843430/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="86377258"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="86377258"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 86377258; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=86377258]").text(description); $(".js-view-count[data-work-id=86377258]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 86377258; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='86377258']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "adef441297e20248975f6c10a132c006" } } $('.js-work-strip[data-work-id=86377258]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":86377258,"title":"a.e. Convergence of Nörlund means with respect to Vilenkin systems of integrable functions","translated_title":"","metadata":{"abstract":"In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise and norm convergence in L_p norms of such Nörlund means.","publication_date":{"day":1,"month":7,"year":2021,"errors":{}}},"translated_abstract":"In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") if (false) { Aedu.setUpFigureCarousel('profile-work-86377258-figures'); } }); </script> <div class="js-work-strip profile--work_container" data-work-id="86377257"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/86377257/On_the_strong_convergence_of_partial_sums_with_respect_to_bounded_Vilenkin_systems"><img alt="Research paper thumbnail of On the strong convergence of partial sums with respect to bounded Vilenkin systems" class="work-thumbnail" src="https://attachments.academia-assets.com/90843462/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377257/On_the_strong_convergence_of_partial_sums_with_respect_to_bounded_Vilenkin_systems">On the strong convergence of partial sums with respect to bounded Vilenkin systems</a></div><div class="wp-workCard_item"><span>arXiv: Classical Analysis and ODEs</span><span>, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we investigate some strong convergence theorems for partial sums with respect to Vi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="6d76064d19bae819feb00b12fc6e2e55" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":90843462,"asset_id":86377257,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/90843462/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="86377257"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="86377257"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 86377257; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") if (false) { Aedu.setUpFigureCarousel('profile-work-86377257-figures'); } }); </script> <div class="js-work-strip profile--work_container" data-work-id="86377256"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" href="https://www.academia.edu/86377256/Lebesgue_and_Vilenkin_Lebesgue_points_and_a_e_Convergence_of_N_orlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions"><img alt="Research paper thumbnail of Lebesgue and Vilenkin-Lebesgue points and a. e. Convergence of N\"orlund means with respect to Vilenkin systems of integrable functions" class="work-thumbnail" src="https://attachments.academia-assets.com/90843431/thumbnails/1.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377256/Lebesgue_and_Vilenkin_Lebesgue_points_and_a_e_Convergence_of_N_orlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions">Lebesgue and Vilenkin-Lebesgue points and a. e. Convergence of N\"orlund means with respect to Vilenkin systems of integrable functions</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coeffi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise convergence and convergence in Lp norms of such Nörlund means. 2000 Mathematics Subject Classification. 42C10, 42B25.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="10a81e4bf46504697ce844564f904220" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":90843431,"asset_id":86377256,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/90843431/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="86377256"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="86377256"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 86377256; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=86377256]").text(description); $(".js-view-count[data-work-id=86377256]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 86377256; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='86377256']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "10a81e4bf46504697ce844564f904220" } } $('.js-work-strip[data-work-id=86377256]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":86377256,"title":"Lebesgue and Vilenkin-Lebesgue points and a. e. Convergence of N\\\"orlund means with respect to Vilenkin systems of integrable functions","translated_title":"","metadata":{"abstract":"In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. Moreover, we discuss pointwise convergence and convergence in Lp norms of such Nörlund means. 2000 Mathematics Subject Classification. 42C10, 42B25.","ai_title_tag":"Nörlund Means Convergence in Vilenkin Systems of Integrable Functions","publication_date":{"day":null,"month":null,"year":2021,"errors":{}}},"translated_abstract":"In this paper we derive converge of Nörlund means of Vilenkin-Fourier series with monotone coefficients of integrable functions in Lebesgue and Vilinkin-Lebesgue points. 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Moreover, we discuss pointwise convergence and convergence in Lp norms of such Nörlund means. 2000 Mathematics Subject Classification. 42C10, 42B25.","owner":{"id":127157369,"first_name":"Giorgi","middle_initials":null,"last_name":"Tutberidze","page_name":"Tutberidze","domain_name":"independent","created_at":"2019-09-19T03:49:23.296-07:00","display_name":"Giorgi 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data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377210/A_note_on_The_maximal_operators_of_the_N_orlund_logaritmic_means_of_Vilenkin_Fourier_series">A note on The maximal operators of the N\"orlund logaritmic means of Vilenkin-Fourier series</a></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The main aim of this paper is to investigate $\left(H_{p},L_{p}\right)$- type inequalities for th...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The main aim of this paper is to investigate $\left(H_{p},L_{p}\right)$- type inequalities for the the maximal operators of N\&quot;orlund logaritmic means, for $0</span></div><div class="wp-workCard_item wp-workCard--actions"><span 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wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/86377209/Some_inequalities_related_to_strong_convergence_of_Riesz_logarithmic_means">Some inequalities related to strong convergence of Riesz logarithmic means</a></div><div class="wp-workCard_item"><span>Journal of Inequalities and Applications</span><span>, 2020</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-di...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin–Fourier (Walsh–Fourier) series. The corresponding inequality is pointed out and it is also proved that the inequality is in a sense sharp, at least for the case with Walsh–Fourier series.</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="a6fb49d11fe86df3894d5687a1fcfae2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":90843386,"asset_id":86377209,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/90843386/download_file?s=profile"><span><i class="fa fa-arrow-down"></i></span><span>Download</span></a><span class="wp-workCard--action visible-if-viewed-by-owner inline-block" style="display: none;"><span class="js-profile-work-strip-edit-button-wrapper profile-work-strip-edit-button-wrapper" data-work-id="86377209"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="86377209"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 86377209; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=86377209]").text(description); $(".js-view-count[data-work-id=86377209]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 86377209; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='86377209']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span></div><div id="work-strip-premium-row-container"></div></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/wow_profile-a9bf3a2bc8c89fa2a77156577594264ee8a0f214d74241bc0fcd3f69f8d107ac.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (true){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "a6fb49d11fe86df3894d5687a1fcfae2" } } $('.js-work-strip[data-work-id=86377209]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":86377209,"title":"Some inequalities related to strong convergence of Riesz logarithmic means","translated_title":"","metadata":{"abstract":"In this paper we derive a new strong convergence theorem of Riesz logarithmic means of the one-dimensional Vilenkin–Fourier (Walsh–Fourier) series. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") if (false) { Aedu.setUpFigureCarousel('profile-work-86377209-figures'); } }); </script> <div class="js-work-strip profile--work_container" data-work-id="71066348"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" rel="nofollow" href="https://www.academia.edu/71066348/Modulus_of_continuity_and_convergence_of_subsequences_of_Vilenkin_Fej%C3%A9r_means_in_martingale_Hardy_spaces"><img alt="Research paper thumbnail of Modulus of continuity and convergence of subsequences of Vilenkin–Fejér means in martingale Hardy spaces" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title">Modulus of continuity and convergence of subsequences of Vilenkin–Fejér means in martingale Hardy spaces</div><div class="wp-workCard_item"><span>Georgian Mathematical Journal</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In this paper, we find a necessary and sufficient condition for the modulus of continuity for whi...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">In this paper, we find a necessary and sufficient condition for the modulus of continuity for which subsequences of Fejér means with respect to Vilenkin systems are bounded from the Hardy space H p {H_{p}} to the Lebesgue space L p {L_{p}} for all 0 &lt; 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") if (false) { Aedu.setUpFigureCarousel('profile-work-71066348-figures'); } }); </script> <div class="js-work-strip profile--work_container" data-work-id="61065297"><div class="profile--work_thumbnail hidden-xs"><a class="js-work-strip-work-link" data-click-track="profile-work-strip-thumbnail" rel="nofollow" href="https://www.academia.edu/61065297/Absolute_convergence_factors_of_Lipschitz_class_functions_for_general_Fourier_series"><img alt="Research paper thumbnail of Absolute convergence factors of Lipschitz class functions for general Fourier series" class="work-thumbnail" src="https://a.academia-assets.com/images/blank-paper.jpg" /></a></div><div class="wp-workCard wp-workCard_itemContainer"><div class="wp-workCard_item wp-workCard--title">Absolute convergence factors of Lipschitz class functions for general Fourier series</div><div class="wp-workCard_item"><span>Georgian Mathematical Journal</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The main aim of this paper is to investigate the sequences of positive numbers, for which multipl...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">The main aim of this paper is to investigate the sequences of positive numbers, for which multiplication with Fourier coefficients of functions f ∈ Lip ⁡ 1 {f\in\operatorname{Lip}1} class provides absolute convergence of Fourier series. In particular, we found special conditions for the functions of orthonormal system (ONS), for which the above sequences are absolute convergence factors of Fourier series of functions of Lip ⁡ 1 {\operatorname{Lip}1} class. 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