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class="social-profile-avatar-container"><img class="profile-avatar u-positionAbsolute" alt="Gyorgy Gat" 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/32476274/18444977/18396189/s200_gyorgy.gat.jpg" /></div><div class="title-container"><h1 class="ds2-5-heading-sans-serif-sm">Gyorgy Gat</h1><div class="affiliations-container fake-truncate js-profile-affiliations"><div><a class="u-tcGrayDarker" href="https://unideb.academia.edu/">University of Debrecen</a>, <a class="u-tcGrayDarker" href="https://unideb.academia.edu/Departments/Institute_of_Mathematics/Documents">Institute of Mathematics</a>, <span class="u-tcGrayDarker">Faculty Member</span></div></div></div></div><div class="sidebar-cta-container"><button class="ds2-5-button hidden profile-cta-button grow js-profile-follow-button" 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class="label">Following</p><p class="data">13</p></div></a><span><div class="stat-container"><p class="label"><span class="js-profile-total-view-text">Public Views</span></p><p class="data"><span class="js-profile-view-count"></span></p></div></span></div><div class="user-bio-container"><div class="profile-bio fake-truncate js-profile-about" style="margin: 0px;">I am a mathematician interested in Fourier analysis of one and multivariable functions mainly with respect to the trigonometric, Walsh and Walsh-like systems.<br /><div class="js-profile-less-about u-linkUnstyled u-tcGrayDarker u-textDecorationUnderline u-displayNone">less</div></div></div><div class="ri-section"><div class="ri-section-header"><span>Interests</span></div><div class="ri-tags-container"><a data-click-track="profile-user-info-expand-research-interests" data-has-card-for-ri-list="32476274" href="https://www.academia.edu/Documents/in/Fourier_Analysis"><div id="js-react-on-rails-context" style="display:none" 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class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110444358/On_almost_everywhere_convergence_of_the_generalized_Marcienkiwicz_means_with_respect_to_two_dimensional_Vilenkin_like_systems">On almost everywhere convergence of the generalized Marcienkiwicz means with respect to two dimensional Vilenkin-like systems</a></div><div class="wp-workCard_item"><span>Miskolc Mathematical Notes</span><span>, 2020</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="c4210aa0f402c0915cfc022b85b007b6" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108260154,"asset_id":110444358,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108260154/download_file?st=MTczMjM5NTg1Nyw4LjIyMi4yMDguMTQ2&s=profile"><span><i class="fa 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Series electronics and energetics</span><span>, 2008</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="155c6df59d0f3f506b512bf16578d9b2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108260151,"asset_id":110444357,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108260151/download_file?st=MTczMjM5NTg1Nyw4LjIyMi4yMDguMTQ2&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="110444357"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110444357"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110444357; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110444357]").text(description); $(".js-view-count[data-work-id=110444357]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 110444357; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110444357']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 110444357, container: "", }); });</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-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.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: "155c6df59d0f3f506b512bf16578d9b2" } } $('.js-work-strip[data-work-id=110444357]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444357,"title":"Almost everywhere convergence of a subsequence of the logarithmic means of Vilenkin-Fourier series","translated_title":"","metadata":{"publisher":"University of Niš","publication_date":{"day":null,"month":null,"year":2008,"errors":{}},"publication_name":"Facta universitatis. 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Matematika</span><span>, Jul 18, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The element of the Walsh system, that is the Walsh functions map from the unit interval to the se...</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 element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. They can be extended to the set of nonnegative reals, but not to the whole real line. The aim of this article is to give an Walsh-like orthonormal and complete function system which can be extended on the real line.</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="110444353"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110444353"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110444353; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110444353]").text(description); $(".js-view-count[data-work-id=110444353]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 110444353; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110444353']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 110444353, container: "", }); });</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-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=110444353]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444353,"title":"The Walsh-Fourier transform on the real line","translated_title":"","metadata":{"abstract":"The element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110444344"><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/110444344/Convergence_of_a_Subsequence_of_Triangular_Partial_Sums_of_Double_Walsh_Fourier_Series"><img alt="Research paper thumbnail of Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-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"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110444344/Convergence_of_a_Subsequence_of_Triangular_Partial_Sums_of_Double_Walsh_Fourier_Series">Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series</a></div><div class="wp-workCard_item"><span>Journal of Contemporary Mathematical Analysis</span><span>, Jul 1, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In 1987 Harris proved-among others that for each 1 ≤ p &lt; 2 there exists a two-dimensional func...</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 1987 Harris proved-among others that for each 1 ≤ p &lt; 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p &lt; 2) with subsequence of triangular partial means $$S_{2^A}^\Delta(f)$$ of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence $$S_{a(n)}^\Delta (f) \rightarrow f$$ holds, where a(n) is a lacunary sequence of positive integers.</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="110444344"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110444344"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110444344; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110444344]").text(description); $(".js-view-count[data-work-id=110444344]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 110444344; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110444344']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 110444344, container: "", }); });</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-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=110444344]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444344,"title":"Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series","translated_title":"","metadata":{"abstract":"In 1987 Harris proved-among others that for each 1 ≤ p \u0026lt; 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p \u0026lt; 2) with subsequence of triangular partial means $$S_{2^A}^\\Delta(f)$$ of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence $$S_{a(n)}^\\Delta (f) \\rightarrow f$$ holds, where a(n) is a lacunary sequence of positive integers.","publisher":"MAIK Nauka/Interperiodica","publication_date":{"day":1,"month":7,"year":2019,"errors":{}},"publication_name":"Journal of Contemporary Mathematical Analysis"},"translated_abstract":"In 1987 Harris proved-among others that for each 1 ≤ p \u0026lt; 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p \u0026lt; 2) with subsequence of triangular partial means $$S_{2^A}^\\Delta(f)$$ of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). 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href="https://www.academia.edu/110444342/Norm_convergence_of_double_Fourier_series_on_unbounded_Vilenkin_groups"><img alt="Research paper thumbnail of Norm convergence of double Fourier series on unbounded Vilenkin groups" class="work-thumbnail" src="https://attachments.academia-assets.com/108260241/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/110444342/Norm_convergence_of_double_Fourier_series_on_unbounded_Vilenkin_groups">Norm convergence of double Fourier series on unbounded Vilenkin groups</a></div><div class="wp-workCard_item"><span>Acta Mathematica Hungarica</span><span>, Mar 20, 2017</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f8d5fdd9b461ed1aa8a5e4d16be3cb57" class="wp-workCard--action" 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/></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/110444324/Almost_everywhere_convergence_of_Riesz_means_of_one_dimensional_Fourier_series_on_the_group_of_2_adic_integers">Almost everywhere convergence of Riesz means of one-dimensional Fourier series on the group of 2-adic integers</a></div><div class="wp-workCard_item"><span>Novi Sad Journal of Mathematics</span><span>, Apr 29, 2021</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="110444324"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa 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$('.js-work-strip[data-work-id=110444324]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444324,"title":"Almost everywhere convergence of Riesz means of one-dimensional Fourier series on the group of 2-adic integers","translated_title":"","metadata":{"publisher":"Faculty of Sciences, University of Novi Sad","publication_date":{"day":29,"month":4,"year":2021,"errors":{}},"publication_name":"Novi Sad Journal of 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Gat","url":"https://unideb.academia.edu/GyorgyGat"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":1228829,"name":"Fourier Series","url":"https://www.academia.edu/Documents/in/Fourier_Series"}],"urls":[{"id":36348739,"url":"https://doi.org/10.30755/nsjom.12069"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110444323"><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/110444323/Almost_Everywhere_Convergence_of_Subsequence_of_Quadratic_Partial_Sums_of_Two_Dimensional_Walsh_Fourier_Series"><img alt="Research paper thumbnail of Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–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"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110444323/Almost_Everywhere_Convergence_of_Subsequence_of_Quadratic_Partial_Sums_of_Two_Dimensional_Walsh_Fourier_Series">Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–Fourier Series</a></div><div class="wp-workCard_item"><span>Analysis Mathematica</span><span>, Mar 1, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): ...</span><a class="js-work-more-abstract" data-broccoli-component="work_strip.more_abstract" data-click-track="profile-work-strip-more-abstract" href="javascript:;"><span> more </span><span><i class="fa fa-caret-down"></i></span></a><span class="js-work-more-abstract-untruncated hidden">For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): = \sum\limits_{j = 0}^\infty {\left| {{n_j} - {n_{j + 1}}} \right| + {n_0},}$$V(n):=∑j=0∞|nj−nj+1|+n0, where n := ∑i=0∞ni2i, ni ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition supAV (nA) &lt; ∞, the subsequence of quadratic partial sums $$S{_n^{\square}}_A\left( f \right)$$Sn□A(f) of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {nA : A ≥ 1} with the condition supAV(nA) &lt; ∞ and a function f ∈ φ(L)(I2) for which $$\text{sup} _A|S{_n^{\square}}_A\left( {{x^1},{x^2};f} \right)| = \infty $$supA|Sn◻A(x1,x2;f)|=∞ for almost all (x1, x2) ∈ I2.</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="110444323"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110444323"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110444323; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110444323]").text(description); $(".js-view-count[data-work-id=110444323]").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 = 110444323; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110444323']"); 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><span><script>$(function() { new Works.PaperRankView({ workId: 110444323, container: "", }); });</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-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=110444323]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444323,"title":"Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–Fourier Series","translated_title":"","metadata":{"abstract":"For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): = \\sum\\limits_{j = 0}^\\infty {\\left| {{n_j} - {n_{j + 1}}} \\right| + {n_0},}$$V(n):=∑j=0∞|nj−nj+1|+n0, where n := ∑i=0∞ni2i, ni ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition supAV (nA) \u0026lt; ∞, the subsequence of quadratic partial sums $$S{_n^{\\square}}_A\\left( f \\right)$$Sn□A(f) of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {nA : A ≥ 1} with the condition supAV(nA) \u0026lt; ∞ and a function f ∈ φ(L)(I2) for which $$\\text{sup} _A|S{_n^{\\square}}_A\\left( {{x^1},{x^2};f} \\right)| = \\infty $$supA|Sn◻A(x1,x2;f)|=∞ for almost all (x1, x2) ∈ I2.","publisher":"Springer Science+Business Media","publication_date":{"day":1,"month":3,"year":2018,"errors":{}},"publication_name":"Analysis Mathematica"},"translated_abstract":"For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): = \\sum\\limits_{j = 0}^\\infty {\\left| {{n_j} - {n_{j + 1}}} \\right| + {n_0},}$$V(n):=∑j=0∞|nj−nj+1|+n0, where n := ∑i=0∞ni2i, ni ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition supAV (nA) \u0026lt; ∞, the subsequence of quadratic partial sums $$S{_n^{\\square}}_A\\left( f \\right)$$Sn□A(f) of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {nA : A ≥ 1} with the condition supAV(nA) \u0026lt; ∞ and a function f ∈ φ(L)(I2) for which $$\\text{sup} _A|S{_n^{\\square}}_A\\left( {{x^1},{x^2};f} \\right)| = \\infty $$supA|Sn◻A(x1,x2;f)|=∞ for almost all (x1, x2) ∈ I2.","internal_url":"https://www.academia.edu/110444323/Almost_Everywhere_Convergence_of_Subsequence_of_Quadratic_Partial_Sums_of_Two_Dimensional_Walsh_Fourier_Series","translated_internal_url":"","created_at":"2023-12-03T06:26:36.775-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":32476274,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Almost_Everywhere_Convergence_of_Subsequence_of_Quadratic_Partial_Sums_of_Two_Dimensional_Walsh_Fourier_Series","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":32476274,"first_name":"Gyorgy","middle_initials":null,"last_name":"Gat","page_name":"GyorgyGat","domain_name":"unideb","created_at":"2015-06-23T14:37:53.954-07:00","display_name":"Gyorgy Gat","url":"https://unideb.academia.edu/GyorgyGat"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":454069,"name":"Subsequence","url":"https://www.academia.edu/Documents/in/Subsequence"},{"id":1228829,"name":"Fourier Series","url":"https://www.academia.edu/Documents/in/Fourier_Series"}],"urls":[{"id":36348738,"url":"https://doi.org/10.1007/s10476-018-0107-2"}]}, dispatcherData: dispatcherData }); 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Series electronics and energetics</span><span>, 2008</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="155c6df59d0f3f506b512bf16578d9b2" class="wp-workCard--action" rel="nofollow" data-click-track="profile-work-strip-download" data-download="{"attachment_id":108260151,"asset_id":110444357,"asset_type":"Work","button_location":"profile"}" href="https://www.academia.edu/attachments/108260151/download_file?st=MTczMjM5NTg1Nyw4LjIyMi4yMDguMTQ2&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="110444357"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110444357"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110444357; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110444357]").text(description); $(".js-view-count[data-work-id=110444357]").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 = 110444357; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110444357']"); 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><span><script>$(function() { new Works.PaperRankView({ workId: 110444357, container: "", }); });</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-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.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: "155c6df59d0f3f506b512bf16578d9b2" } } $('.js-work-strip[data-work-id=110444357]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444357,"title":"Almost everywhere convergence of a subsequence of the logarithmic means of Vilenkin-Fourier series","translated_title":"","metadata":{"publisher":"University of Niš","publication_date":{"day":null,"month":null,"year":2008,"errors":{}},"publication_name":"Facta universitatis. 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Matematika</span><span>, Jul 18, 2022</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">The element of the Walsh system, that is the Walsh functions map from the unit interval to the se...</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 element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. They can be extended to the set of nonnegative reals, but not to the whole real line. The aim of this article is to give an Walsh-like orthonormal and complete function system which can be extended on the real line.</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="110444353"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110444353"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110444353; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110444353]").text(description); $(".js-view-count[data-work-id=110444353]").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 = 110444353; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110444353']"); 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><span><script>$(function() { new Works.PaperRankView({ workId: 110444353, container: "", }); });</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-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=110444353]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444353,"title":"The Walsh-Fourier transform on the real line","translated_title":"","metadata":{"abstract":"The element of the Walsh system, that is the Walsh functions map from the unit interval to the set {−1, 1}. 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$(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110444344"><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/110444344/Convergence_of_a_Subsequence_of_Triangular_Partial_Sums_of_Double_Walsh_Fourier_Series"><img alt="Research paper thumbnail of Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-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"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110444344/Convergence_of_a_Subsequence_of_Triangular_Partial_Sums_of_Double_Walsh_Fourier_Series">Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series</a></div><div class="wp-workCard_item"><span>Journal of Contemporary Mathematical Analysis</span><span>, Jul 1, 2019</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">In 1987 Harris proved-among others that for each 1 ≤ p &lt; 2 there exists a two-dimensional func...</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 1987 Harris proved-among others that for each 1 ≤ p &lt; 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p &lt; 2) with subsequence of triangular partial means $$S_{2^A}^\Delta(f)$$ of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence $$S_{a(n)}^\Delta (f) \rightarrow f$$ holds, where a(n) is a lacunary sequence of positive integers.</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="110444344"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110444344"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110444344; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110444344]").text(description); $(".js-view-count[data-work-id=110444344]").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 = 110444344; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110444344']"); 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><span><script>$(function() { new Works.PaperRankView({ workId: 110444344, container: "", }); });</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-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=110444344]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444344,"title":"Convergence of a Subsequence of Triangular Partial Sums of Double Walsh-Fourier Series","translated_title":"","metadata":{"abstract":"In 1987 Harris proved-among others that for each 1 ≤ p \u0026lt; 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p \u0026lt; 2) with subsequence of triangular partial means $$S_{2^A}^\\Delta(f)$$ of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). We also prove that for each function f ∈ L2(II2) a.e. convergence $$S_{a(n)}^\\Delta (f) \\rightarrow f$$ holds, where a(n) is a lacunary sequence of positive integers.","publisher":"MAIK Nauka/Interperiodica","publication_date":{"day":1,"month":7,"year":2019,"errors":{}},"publication_name":"Journal of Contemporary Mathematical Analysis"},"translated_abstract":"In 1987 Harris proved-among others that for each 1 ≤ p \u0026lt; 2 there exists a two-dimensional function f ∈ Lp such that its triangular Walsh-Fourier series does not converge almost everywhere. In this paper we prove that the set of the functions from the space Lp(II2) (1 ≤ p \u0026lt; 2) with subsequence of triangular partial means $$S_{2^A}^\\Delta(f)$$ of the double Walsh-Fourier series convergent in measure on II2 is of first Baire category in Lp(II2). 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href="https://www.academia.edu/110444342/Norm_convergence_of_double_Fourier_series_on_unbounded_Vilenkin_groups"><img alt="Research paper thumbnail of Norm convergence of double Fourier series on unbounded Vilenkin groups" class="work-thumbnail" src="https://attachments.academia-assets.com/108260241/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/110444342/Norm_convergence_of_double_Fourier_series_on_unbounded_Vilenkin_groups">Norm convergence of double Fourier series on unbounded Vilenkin groups</a></div><div class="wp-workCard_item"><span>Acta Mathematica Hungarica</span><span>, Mar 20, 2017</span></div><div class="wp-workCard_item wp-workCard--actions"><span class="work-strip-bookmark-button-container"></span><a id="f8d5fdd9b461ed1aa8a5e4d16be3cb57" class="wp-workCard--action" 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/></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/110444324/Almost_everywhere_convergence_of_Riesz_means_of_one_dimensional_Fourier_series_on_the_group_of_2_adic_integers">Almost everywhere convergence of Riesz means of one-dimensional Fourier series on the group of 2-adic integers</a></div><div class="wp-workCard_item"><span>Novi Sad Journal of Mathematics</span><span>, Apr 29, 2021</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="110444324"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa 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$('.js-work-strip[data-work-id=110444324]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444324,"title":"Almost everywhere convergence of Riesz means of one-dimensional Fourier series on the group of 2-adic integers","translated_title":"","metadata":{"publisher":"Faculty of Sciences, University of Novi Sad","publication_date":{"day":29,"month":4,"year":2021,"errors":{}},"publication_name":"Novi Sad Journal of 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Gat","url":"https://unideb.academia.edu/GyorgyGat"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":1228829,"name":"Fourier Series","url":"https://www.academia.edu/Documents/in/Fourier_Series"}],"urls":[{"id":36348739,"url":"https://doi.org/10.30755/nsjom.12069"}]}, dispatcherData: dispatcherData }); $(this).data('initialized', true); } }); $a.trackClickSource(".js-work-strip-work-link", "profile_work_strip") }); </script> <div class="js-work-strip profile--work_container" data-work-id="110444323"><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/110444323/Almost_Everywhere_Convergence_of_Subsequence_of_Quadratic_Partial_Sums_of_Two_Dimensional_Walsh_Fourier_Series"><img alt="Research paper thumbnail of Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–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"><a class="js-work-strip-work-link text-gray-darker" data-click-track="profile-work-strip-title" href="https://www.academia.edu/110444323/Almost_Everywhere_Convergence_of_Subsequence_of_Quadratic_Partial_Sums_of_Two_Dimensional_Walsh_Fourier_Series">Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–Fourier Series</a></div><div class="wp-workCard_item"><span>Analysis Mathematica</span><span>, Mar 1, 2018</span></div><div class="wp-workCard_item"><span class="js-work-more-abstract-truncated">For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): ...</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">For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): = \sum\limits_{j = 0}^\infty {\left| {{n_j} - {n_{j + 1}}} \right| + {n_0},}$$V(n):=∑j=0∞|nj−nj+1|+n0, where n := ∑i=0∞ni2i, ni ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition supAV (nA) &lt; ∞, the subsequence of quadratic partial sums $$S{_n^{\square}}_A\left( f \right)$$Sn□A(f) of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {nA : A ≥ 1} with the condition supAV(nA) &lt; ∞ and a function f ∈ φ(L)(I2) for which $$\text{sup} _A|S{_n^{\square}}_A\left( {{x^1},{x^2};f} \right)| = \infty $$supA|Sn◻A(x1,x2;f)|=∞ for almost all (x1, x2) ∈ I2.</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="110444323"><a class="js-profile-work-strip-edit-button" tabindex="0"><span><i class="fa fa-pencil"></i></span><span>Edit</span></a></span></span><span id="work-strip-rankings-button-container"></span></div><div class="wp-workCard_item wp-workCard--stats"><span><span><span class="js-view-count view-count u-mr2x" data-work-id="110444323"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 110444323; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=110444323]").text(description); $(".js-view-count[data-work-id=110444323]").attr('title', description).tooltip(); }); });</script></span></span><span><span class="percentile-widget hidden"><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 110444323; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-work-strip[data-work-id='110444323']"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></span><span><script>$(function() { new Works.PaperRankView({ workId: 110444323, container: "", }); });</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-f77ea15d77ce96025a6048a514272ad8becbad23c641fc2b3bd6e24ca6ff1932.js","https://a.academia-assets.com/assets/work_edit-ad038b8c047c1a8d4fa01b402d530ff93c45fee2137a149a4a5398bc8ad67560.js"], function() { // from javascript_helper.rb var dispatcherData = {} if (false){ window.WowProfile.dispatcher = window.WowProfile.dispatcher || _.clone(Backbone.Events); dispatcherData = { dispatcher: window.WowProfile.dispatcher, downloadLinkId: "-1" } } $('.js-work-strip[data-work-id=110444323]').each(function() { if (!$(this).data('initialized')) { new WowProfile.WorkStripView({ el: this, workJSON: {"id":110444323,"title":"Almost Everywhere Convergence of Subsequence of Quadratic Partial Sums of Two-Dimensional Walsh–Fourier Series","translated_title":"","metadata":{"abstract":"For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): = \\sum\\limits_{j = 0}^\\infty {\\left| {{n_j} - {n_{j + 1}}} \\right| + {n_0},}$$V(n):=∑j=0∞|nj−nj+1|+n0, where n := ∑i=0∞ni2i, ni ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition supAV (nA) \u0026lt; ∞, the subsequence of quadratic partial sums $$S{_n^{\\square}}_A\\left( f \\right)$$Sn□A(f) of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {nA : A ≥ 1} with the condition supAV(nA) \u0026lt; ∞ and a function f ∈ φ(L)(I2) for which $$\\text{sup} _A|S{_n^{\\square}}_A\\left( {{x^1},{x^2};f} \\right)| = \\infty $$supA|Sn◻A(x1,x2;f)|=∞ for almost all (x1, x2) ∈ I2.","publisher":"Springer Science+Business Media","publication_date":{"day":1,"month":3,"year":2018,"errors":{}},"publication_name":"Analysis Mathematica"},"translated_abstract":"For a non-negative integer n let us denote the dyadic variation of a natural number n by $$V(n): = \\sum\\limits_{j = 0}^\\infty {\\left| {{n_j} - {n_{j + 1}}} \\right| + {n_0},}$$V(n):=∑j=0∞|nj−nj+1|+n0, where n := ∑i=0∞ni2i, ni ∈ {0, 1}. In this paper we prove that for a function f ∈ L log L(I2) under the condition supAV (nA) \u0026lt; ∞, the subsequence of quadratic partial sums $$S{_n^{\\square}}_A\\left( f \\right)$$Sn□A(f) of two-dimensional Walsh–Fourier series converges to the function f almost everywhere. We also prove sharpness of this result. Namely, we prove that for all monotone increasing function φ: [0,∞) → [0,∞) such that φ(u) = o(u log u) as u → ∞ there exists a sequence {nA : A ≥ 1} with the condition supAV(nA) \u0026lt; ∞ and a function f ∈ φ(L)(I2) for which $$\\text{sup} _A|S{_n^{\\square}}_A\\left( {{x^1},{x^2};f} \\right)| = \\infty $$supA|Sn◻A(x1,x2;f)|=∞ for almost all (x1, x2) ∈ I2.","internal_url":"https://www.academia.edu/110444323/Almost_Everywhere_Convergence_of_Subsequence_of_Quadratic_Partial_Sums_of_Two_Dimensional_Walsh_Fourier_Series","translated_internal_url":"","created_at":"2023-12-03T06:26:36.775-08:00","preview_url":null,"current_user_can_edit":null,"current_user_is_owner":null,"owner_id":32476274,"coauthors_can_edit":true,"document_type":"paper","co_author_tags":[],"downloadable_attachments":[],"slug":"Almost_Everywhere_Convergence_of_Subsequence_of_Quadratic_Partial_Sums_of_Two_Dimensional_Walsh_Fourier_Series","translated_slug":"","page_count":null,"language":"en","content_type":"Work","owner":{"id":32476274,"first_name":"Gyorgy","middle_initials":null,"last_name":"Gat","page_name":"GyorgyGat","domain_name":"unideb","created_at":"2015-06-23T14:37:53.954-07:00","display_name":"Gyorgy Gat","url":"https://unideb.academia.edu/GyorgyGat"},"attachments":[],"research_interests":[{"id":300,"name":"Mathematics","url":"https://www.academia.edu/Documents/in/Mathematics"},{"id":5436,"name":"Combinatorics","url":"https://www.academia.edu/Documents/in/Combinatorics"},{"id":19997,"name":"Pure Mathematics","url":"https://www.academia.edu/Documents/in/Pure_Mathematics"},{"id":454069,"name":"Subsequence","url":"https://www.academia.edu/Documents/in/Subsequence"},{"id":1228829,"name":"Fourier Series","url":"https://www.academia.edu/Documents/in/Fourier_Series"}],"urls":[{"id":36348738,"url":"https://doi.org/10.1007/s10476-018-0107-2"}]}, dispatcherData: dispatcherData }); 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