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In other words, we prove that for any Orlicz space, which is not a subspace of L log L(I 2), the set of" /> <meta name="twitter:image" content="https://0.academia-photos.com/32476274/18444977/18396189/s200_gyorgy.gat.jpg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/68924008/Convergence_in_measure_of_logarithmic_means_of_quadratical_partial_sums_of_double_Walsh_Fourier_series" /> <meta property="og:title" content="Convergence in measure of logarithmic means of quadratical partial sums of double Walsh–Fourier series" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="The main aim of this paper is to prove that the logarithmic means of the double Walsh-Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of L log L(I 2), the set of" /> <meta property="article:author" content="https://unideb.academia.edu/GyorgyGat" /> <meta name="description" content="The main aim of this paper is to prove that the logarithmic means of the double Walsh-Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of L log L(I 2), the set of" /> <title>(PDF) Convergence in measure of logarithmic means of quadratical partial sums of double Walsh–Fourier series</title> <link rel="canonical" href="https://www.academia.edu/68924008/Convergence_in_measure_of_logarithmic_means_of_quadratical_partial_sums_of_double_Walsh_Fourier_series" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two 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false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":68924008,"created_at":"2022-01-20T22:05:58.307-08:00","from_world_paper_id":192735739,"updated_at":"2024-11-25T03:18:22.793-08:00","_data":{"grobid_abstract":"The main aim of this paper is to prove that the logarithmic means of the double Walsh-Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of L log L(I 2), the set of functions for which quadratic logarithmic means of the double Walsh-Fourier series converge in measure is of first Baire category.","publication_date":"2006,,","grobid_abstract_attachment_id":"79223354"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Convergence in measure of logarithmic means of quadratical partial sums of double Walsh–Fourier series","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [32476274]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":79223354,"attachmentType":"pdf"}"><img alt="First page of “Convergence in measure of logarithmic means of quadratical partial sums of double Walsh–Fourier series”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/79223354/mini_magick20220120-19689-1sei3u1.png?1642745690" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">Convergence in measure of logarithmic means of quadratical partial sums of double Walsh–Fourier series</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat"><img alt="Profile image of Gyorgy Gat" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/32476274/18444977/18396189/s65_gyorgy.gat.jpg" />Gyorgy Gat</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2006</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">12 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 68924008; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=68924008"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">The main aim of this paper is to prove that the logarithmic means of the double Walsh-Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of L log L(I 2), the set of functions for which quadratic logarithmic means of the double Walsh-Fourier series converge in measure is of first Baire category.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":79223354,"attachmentType":"pdf","workUrl":"https://www.academia.edu/68924008/Convergence_in_measure_of_logarithmic_means_of_quadratical_partial_sums_of_double_Walsh_Fourier_series"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":79223354,"attachmentType":"pdf","workUrl":"https://www.academia.edu/68924008/Convergence_in_measure_of_logarithmic_means_of_quadratical_partial_sums_of_double_Walsh_Fourier_series"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="79223354" data-landing_url="https://www.academia.edu/68924008/Convergence_in_measure_of_logarithmic_means_of_quadratical_partial_sums_of_double_Walsh_Fourier_series" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="68924055" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68924055/Convergence_in_measure_of_logarithmic_means_of_multiple_Fourier_series">Convergence in measure of logarithmic means of multiple Fourier series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Contemporary Mathematical Analysis, 2014</p><p class="ds-related-work--abstract ds2-5-body-sm">The main aim of this paper is to prove that the logarithmic means of the double Walsh-Fourier series do not improve the convergence in measure. In other words, we prove that for any Orlicz space, which is not a subspace of L log L(I 2), the set of functions for which quadratic logarithmic means of the double Walsh-Fourier series converge in measure is of first Baire category.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Convergence in measure of logarithmic means of multiple Fourier series","attachmentId":79223357,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924055/Convergence_in_measure_of_logarithmic_means_of_multiple_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/68924055/Convergence_in_measure_of_logarithmic_means_of_multiple_Fourier_series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="68923986" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68923986/Convergence_of_logarithmic_means_of_multiple_Walsh_Fourier_series">Convergence of logarithmic means of multiple Walsh-Fourier series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Analysis in Theory and Applications, 2005</p><p class="ds-related-work--abstract ds2-5-body-sm">Nörlund logarithmic means of multiple Walsh-Fourier series acting from spaceLlnd-1 L([0, 1)d), d&gt;-1 into space weak-L1([0, 1)d) are studied. The maximal Orlicz space such that the Nörlund logarithmic means of multiple Walsh-Fourier series for the functions from this space converge in d-dimensional measure is found.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Convergence of logarithmic means of multiple Walsh-Fourier series","attachmentId":79222978,"attachmentType":"pdf","work_url":"https://www.academia.edu/68923986/Convergence_of_logarithmic_means_of_multiple_Walsh_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/68923986/Convergence_of_logarithmic_means_of_multiple_Walsh_Fourier_series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="68924097" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68924097/Almost_everywhere_convergence_of_a_subsequence_of_the_logarithmic_means_of_quadratical_partial_sums_of_double_Walsh_Fourier_series">Almost everywhere convergence of a subsequence of the logarithmic means of quadratical partial sums of double Walsh-Fourier series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Publicationes Mathematicae, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">The main aim of this paper is to prove that the maximal operator of the logarithmic means of quadratical partial sums of double Walsh-Fourier series is of weak type (1, 1) provided that the supremum in the maximal operator is taken over special indicies. The set of Walsh polynomials is dense in L 1 (I × I) , so by the well-known density argument we have that t 2 n f x 1 , x 2 → f x 1 , x 2 a. e. for all integrable two-variable function f .</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Almost everywhere convergence of a subsequence of the logarithmic means of quadratical partial sums of double Walsh-Fourier series","attachmentId":79223359,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924097/Almost_everywhere_convergence_of_a_subsequence_of_the_logarithmic_means_of_quadratical_partial_sums_of_double_Walsh_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/68924097/Almost_everywhere_convergence_of_a_subsequence_of_the_logarithmic_means_of_quadratical_partial_sums_of_double_Walsh_Fourier_series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="68923982" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68923982/Uniform_and_L_convergence_of_Logarithmic_Means_of_Walsh_Fourier_Series">Uniform and L–convergence of Logarithmic Means of Walsh–Fourier Series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Acta Mathematica Sinica, English Series, 2006</p><p class="ds-related-work--abstract ds2-5-body-sm">We discuss some convergence and divergence properties of twodimensional (Nörlund) logarithmic means of two-dimensional Walsh-Fourier series of functions both in the uniform and in the Lebesgue norm. We give necessary and sufficient conditions for the convergence regarding the modulus of continuity of the function, and also the function space.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Uniform and L–convergence of Logarithmic Means of Walsh–Fourier Series","attachmentId":79223055,"attachmentType":"pdf","work_url":"https://www.academia.edu/68923982/Uniform_and_L_convergence_of_Logarithmic_Means_of_Walsh_Fourier_Series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/68923982/Uniform_and_L_convergence_of_Logarithmic_Means_of_Walsh_Fourier_Series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="68924061" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68924061/Uniform_and_L_Convergence_of_Logarithmic_Means_of_Double_Walsh_Fourier_Series">Uniform and 𝐿-Convergence of Logarithmic Means of Double Walsh–Fourier Series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--abstract ds2-5-body-sm">We discuss some convergence and divergence properties of twodimensional (Nörlund) logarithmic means of two-dimensional Walsh-Fourier series of functions both in the uniform and in the Lebesgue norm. We give necessary and sufficient conditions for the convergence regarding the modulus of continuity of the function, and also the function space.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Uniform and 𝐿-Convergence of Logarithmic Means of Double Walsh–Fourier Series","attachmentId":79223039,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924061/Uniform_and_L_Convergence_of_Logarithmic_Means_of_Double_Walsh_Fourier_Series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/68924061/Uniform_and_L_Convergence_of_Logarithmic_Means_of_Double_Walsh_Fourier_Series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="68924074" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68924074/Almost_everywhere_strong_summability_of_Marcinkiewicz_means_of_double_Walsh_Fourier_series">Almost everywhere strong summability of Marcinkiewicz means of double Walsh-Fourier series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Analysis Mathematica, 2014</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation (1 n n−1 m=0 |Smmf − f | p) 1/p → 0 for every two-dimensional functions belonging to L log L and 0 < p ≤ 2. From the theorem of Getsadze [6] it follows that the space L log L can not be enlarged with preserving this strong summability property.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Almost everywhere strong summability of Marcinkiewicz means of double Walsh-Fourier series","attachmentId":79223376,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924074/Almost_everywhere_strong_summability_of_Marcinkiewicz_means_of_double_Walsh_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/68924074/Almost_everywhere_strong_summability_of_Marcinkiewicz_means_of_double_Walsh_Fourier_series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="68924094" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68924094/Uniform_and_L_convergence_of_logarithmic_means_of_cubical_partial_sums_of_double_Walsh_Fourier_series">Uniform and $L$-convergence of logarithmic means of cubical partial sums of double Walsh-Fourier series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2004</p><p class="ds-related-work--abstract ds2-5-body-sm">We discuss some convergence and divergence properties of twodimensional (Nörlund) logarithmic means of two-dimensional Walsh-Fourier series of functions both in the uniform and in the Lebesgue norm. We give necessary and sufficient conditions for the convergence regarding the modulus of continuity of the function, and also the function space.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Uniform and $L$-convergence of logarithmic means of cubical partial sums of double Walsh-Fourier series","attachmentId":79223452,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924094/Uniform_and_L_convergence_of_logarithmic_means_of_cubical_partial_sums_of_double_Walsh_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/68924094/Uniform_and_L_convergence_of_logarithmic_means_of_cubical_partial_sums_of_double_Walsh_Fourier_series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="68924106" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68924106/Maximal_Convergence_Space_of_a_Subsequence_of_the_Logarithmic_Means_of_Rectangular_Partial_Sums_of_Double_Walsh_Fourier_Series">Maximal Convergence Space of a Subsequence of the Logarithmic Means of Rectangular Partial Sums of Double Walsh-Fourier Series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Real Analysis Exchange</p><p class="ds-related-work--abstract ds2-5-body-sm">The main aim of this paper is to prove that the maximal operator of the logarithmic means of rectangular partial sums of double Walsh-Fourier series is of type (H # , L 1) provided that the supremum in the maximal operator is taken over some special indicies. The set of Walsh polynomials is dense in H # , so by the well-known density argument we have that t 2 n ,2 m f x 1 , x 2 → f x 1 , x 2 a. e. as m, n → ∞ for all f ∈ H # (⊃ L log + L). We also prove the sharpness of this result. Namely, For all measurable function δ : [0, +∞) → [0, +∞), lim t→∞ δ(t) = 0 we have a function f such as f ∈ Llog + Lδ(L) and the two-dimensional Nörlund logarithmic means does not converge to f a.e. (in the Pringsheim sense) on I 2 .</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Maximal Convergence Space of a Subsequence of the Logarithmic Means of Rectangular Partial Sums of Double Walsh-Fourier Series","attachmentId":79223372,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924106/Maximal_Convergence_Space_of_a_Subsequence_of_the_Logarithmic_Means_of_Rectangular_Partial_Sums_of_Double_Walsh_Fourier_Series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/68924106/Maximal_Convergence_Space_of_a_Subsequence_of_the_Logarithmic_Means_of_Rectangular_Partial_Sums_of_Double_Walsh_Fourier_Series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="110444322" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/110444322/Almost_everywhere_strong_summability_of_double_Walsh_Fourier_series">Almost everywhere strong summability of double Walsh-Fourier series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Contemporary Mathematical Analysis, 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we study the a. e. strong convergence of the quadratical partial sums of the two-dimensional Walsh-Fourier series. Namely, we prove the a.e. relation (1 n n−1 m=0 |Smmf − f | p) 1/p → 0 for every two-dimensional functions belonging to L log L and 0 < p ≤ 2. From the theorem of Getsadze [6] it follows that the space L log L can not be enlarged with preserving this strong summability property.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Almost everywhere strong summability of double Walsh-Fourier series","attachmentId":108260131,"attachmentType":"pdf","work_url":"https://www.academia.edu/110444322/Almost_everywhere_strong_summability_of_double_Walsh_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/110444322/Almost_everywhere_strong_summability_of_double_Walsh_Fourier_series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="68923983" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68923983/On_the_divergence_of_N%C3%B6rlund_logarithmic_means_of_Walsh_Fourier_series">On the divergence of Nörlund logarithmic means of Walsh-Fourier series</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Acta Mathematica Sinica, English Series, 2009</p><p class="ds-related-work--abstract ds2-5-body-sm">It is well known in the literature that the logarithmic means 1 log n n−1 k=1 S k (f) k of Walsh or trigonometric Fourier series converge a.e. to the function for each integrable function on the unit interval. This is not the case if we take the partial sums. In this paper we prove that the behavior of the so-called Nörlund logarithmic means 1 log n n−1 k=1 S k (f) n − k is closer to the properties of partial sums in this point of view.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the divergence of Nörlund logarithmic means of Walsh-Fourier series","attachmentId":79223051,"attachmentType":"pdf","work_url":"https://www.academia.edu/68923983/On_the_divergence_of_N%C3%B6rlund_logarithmic_means_of_Walsh_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/68923983/On_the_divergence_of_N%C3%B6rlund_logarithmic_means_of_Walsh_Fourier_series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":79223354,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":79223354,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_79223354" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="122045480" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/122045480/Walsh_Fourier_Series_and_Their_Generalizations_in_Orlicz_Spaces">Walsh–Fourier Series and Their Generalizations in Orlicz Spaces</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="230893575" href="https://independent.academia.edu/CatherineFinet">Catherine Finet</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Mathematical Analysis and Applications, 1998</p><div 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ds2-5-body-link" href="https://www.academia.edu/68924085/A_remark_on_the_divergence_of_strong_power_means_of_Walsh_Fourier_series">A remark on the divergence of strong power means of Walsh-Fourier series</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical Notes, 2014</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A remark on the divergence of strong power means of Walsh-Fourier series","attachmentId":79223443,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924085/A_remark_on_the_divergence_of_strong_power_means_of_Walsh_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" 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href="https://independent.academia.edu/Istv%C3%A1nBlahota">István Blahota</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Georgian Mathematical Journal, 2012</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Almost everywhere convergence of a subsequence of logarithmic means of Fourier series on the group of 2-adic integers","attachmentId":86521090,"attachmentType":"pdf","work_url":"https://www.academia.edu/79990993/Almost_everywhere_convergence_of_a_subsequence_of_logarithmic_means_of_Fourier_series_on_the_group_of_2_adic_integers","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/79990993/Almost_everywhere_convergence_of_a_subsequence_of_logarithmic_means_of_Fourier_series_on_the_group_of_2_adic_integers"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="6" data-entity-id="68924007" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68924007/Divergence_of_the_C_1_means_of_d_dimensional_Walsh_Fourier_series">Divergence of the (C, 1) means of d-dimensional Walsh-Fourier series</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2001</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Divergence of the (C, 1) means of d-dimensional Walsh-Fourier series","attachmentId":79223200,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924007/Divergence_of_the_C_1_means_of_d_dimensional_Walsh_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/68924007/Divergence_of_the_C_1_means_of_d_dimensional_Walsh_Fourier_series"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" 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data-entity-id="110444357" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/110444357/Almost_everywhere_convergence_of_a_subsequence_of_the_logarithmic_means_of_Vilenkin_Fourier_series">Almost everywhere convergence of a subsequence of the logarithmic means of Vilenkin-Fourier series</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Facta universitatis. Series electronics and energetics, 2008</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Almost everywhere convergence of a subsequence of the logarithmic means of Vilenkin-Fourier series","attachmentId":108260151,"attachmentType":"pdf","work_url":"https://www.academia.edu/110444357/Almost_everywhere_convergence_of_a_subsequence_of_the_logarithmic_means_of_Vilenkin_Fourier_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/110444357/Almost_everywhere_convergence_of_a_subsequence_of_the_logarithmic_means_of_Vilenkin_Fourier_series"><span class="ds2-5-text-link__content">View 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data-entity-id="6119563" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/6119563/Strong_means_and_the_oscillation_of_multiple_Fourier_Walsh_series">Strong means and the oscillation of multiple Fourier-Walsh series</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="9266195" href="https://bakustate.academia.edu/VladimirRodin">Vladimir Rodin</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical Notes, 1994</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Strong means and the oscillation of multiple Fourier-Walsh 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href="https://www.academia.edu/68924006/On_the_Marcinkiewicz_Fej%C3%A9r_means_of_double_Fourier_series_with_respect_to_the_Walsh_Kaczmarz_system">On the Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2009</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz 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href="https://www.academia.edu/110444355/Almost_everywhere_convergence_and_divergence_of_Ces%C3%A0ro_means_with_varying_parameters_of_Walsh_Fourier_series">Almost everywhere convergence and divergence of Cesàro means with varying parameters of Walsh–Fourier series</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Arabian Journal of Mathematics, 2021</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Almost everywhere convergence and divergence of Cesàro means with varying parameters of Walsh–Fourier 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class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/68924048/On_almost_everywhere_convergence_and_divergence_of_Marcinkiewicz_like_means_of_integrable_functions_with_respect_to_the_two_dimensional_Walsh_system">On almost everywhere convergence and divergence of Marcinkiewicz-like means of integrable functions with respect to the two-dimensional Walsh system</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32476274" href="https://unideb.academia.edu/GyorgyGat">Gyorgy Gat</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Approximation Theory, 2012</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On almost everywhere convergence and divergence of Marcinkiewicz-like means of integrable functions with respect to the two-dimensional Walsh system","attachmentId":79223064,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924048/On_almost_everywhere_convergence_and_divergence_of_Marcinkiewicz_like_means_of_integrable_functions_with_respect_to_the_two_dimensional_Walsh_system","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/68924048/On_almost_everywhere_convergence_and_divergence_of_Marcinkiewicz_like_means_of_integrable_functions_with_respect_to_the_two_dimensional_Walsh_system"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="16" data-entity-id="21953862" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/21953862/Cesaro_Summability_of_Double_Walsh_Fourier_Series">Cesaro Summability of Double Walsh-Fourier Series</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="43203026" href="https://independent.academia.edu/FerencSchipp">Ferenc Schipp</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Transactions of the American Mathematical Society, 1992</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Cesaro Summability of Double Walsh-Fourier 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