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In 1999 the author proved that if f AL p ðG m Þ; where p41; then s n f-f almost everywhere. This was the very first ''positive'' result with respect to the a.e. convergence of the Feje´r means of functions on unbounded Vilenkin groups. One of the main difficulties is that the sequence of the L 1 norm of the Feje´r kernels is not bounded. This is a sharp contrast between the unbounded and the bounded Vilenkin systems. The aim of this paper is to discuss the L 1 case. We prove for f AL 1 ðG m Þ that the relation s Mn f-f holds a.e. (M n is the nth generalized power).","publication_date":"2003,,","grobid_abstract_attachment_id":"79223054"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"Cesāro means of integrable functions with respect to unbounded Vilenkin systems","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 = "full_page_mobile_sutd_modal"; 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":79223054,"attachmentType":"pdf"}"><img alt="First page of “Cesāro means of integrable functions with respect to unbounded Vilenkin systems”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/79223054/mini_magick20220120-5937-1snaspm.png?1642745576" /><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">Cesāro means of integrable functions with respect to unbounded Vilenkin systems</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">2003</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">19 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 = 68924005; 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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">One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Feje´r (or ðC; 1Þ) means of functions on unbounded Vilenkin groups. In 1999 the author proved that if f AL p ðG m Þ; where p41; then s n f-f almost everywhere. This was the very first ''positive'' result with respect to the a.e. convergence of the Feje´r means of functions on unbounded Vilenkin groups. One of the main difficulties is that the sequence of the L 1 norm of the Feje´r kernels is not bounded. This is a sharp contrast between the unbounded and the bounded Vilenkin systems. The aim of this paper is to discuss the L 1 case. We prove for f AL 1 ðG m Þ that the relation s Mn f-f holds a.e. (M n is the nth generalized power).</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":79223054,"attachmentType":"pdf","workUrl":"https://www.academia.edu/68924005/Ces%C4%81ro_means_of_integrable_functions_with_respect_to_unbounded_Vilenkin_systems"}">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":79223054,"attachmentType":"pdf","workUrl":"https://www.academia.edu/68924005/Ces%C4%81ro_means_of_integrable_functions_with_respect_to_unbounded_Vilenkin_systems"}"><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="79223054" data-landing_url="https://www.academia.edu/68924005/Ces%C4%81ro_means_of_integrable_functions_with_respect_to_unbounded_Vilenkin_systems" 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="68924043" 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/68924043/Convergence_of_Cesaro_means_of_functions_with_respect_to_unbounded_Vilenkin_systems">Convergence of Cesaro means of functions with respect to unbounded Vilenkin systems</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">One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Fejér (or (C, 1)) means of functions on unbounded Vilenkin groups. The aim of this paper is to give a résumé of the recent developments concerning this matter. Above all, we prove that the maximal operator sup |σ Mn | is of type (H, L 1) on unbounded Vilenkin groups. First, we give a brief introduction to the theory of Vilenkin systems. These orthonormal systems were introduced by N. Ja. Vilenkin in 1947 (see e.g. [25, 1]) as follows. Let m := (m k , k ∈ N) (N := {0, 1,. .. }, P := N \ {0}) be a sequence of integers each of them not less than 2. Let Z m k denote the discrete cyclic group of order m k. That is, Z m k can be represented by the set {0, 1, ..., m k − 1}, with the group operation mod m k addition. Since the groups is discrete, then every subset is open. The normalized Haar measure on Z m k , µ k is dened by µ k ({j}) := 1/m k (j ∈ {0, 1, ..., m k − 1}). Let G m := ∞ × k=0 Z m k. Then every x ∈ G m can be represented by a sequence x = (x i , i ∈ N) , where x i ∈ Z m i (i ∈ N). The group operation on G m (denoted by +) is the coordinate-wise addition (the inverse operation is denoted by −), the measure (denoted by µ), which is the normalized Haar measure, and the topology are the product measure and topology. Consequently, G m is a compact Abelian group. If sup n∈N m n < ∞, then we call G m a bounded Vilenkin group. If the generating sequence m is not bounded, then G m is said to be an unbounded Vilenkin group. The Vilenkin group is metrizable in the following way: d(x, y) := ∞ i=0 |x i − y i | M i+1 (x, y ∈ G m). The topology induced by this metric, the product topology, and the topology given by intervals dened below, are the same. A base for the neighborhoods of G m can be given by the intervals: I 0 (x) := G m , I n (x) := {y = (y i , i ∈ N) ∈ G m : y i = x i for i < n}</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 Cesaro means of functions with respect to unbounded Vilenkin systems","attachmentId":79223528,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924043/Convergence_of_Cesaro_means_of_functions_with_respect_to_unbounded_Vilenkin_systems","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/68924043/Convergence_of_Cesaro_means_of_functions_with_respect_to_unbounded_Vilenkin_systems"><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="68924009" 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/68924009/On_the_pointwise_convergence_of_Cesaro_means_of_two_variable_functions_with_respect_to_unbounded_Vilenkin_systems">On the pointwise convergence of Cesaro means of two-variable functions with respect to unbounded Vilenkin systems</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">One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Fejér (or (C, 1)) means of functions on unbounded Vilenkin groups. There was no known positive result before the author's paper appeared in 1999 [11] with respect to the a.e. convergence of the one-dimensional (C, 1) means of L p (p > 1) functions. This paper is concerned with the almost everywhere convergence of a subsequence of the two-dimensional Fejér means of functions in L log + L. Namely, we prove the a.e. relation lim n,k→∞ σ Mn,M k f = f (for the indices the condition |n − k| < α is provided, where α > 0 is some constant).</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 pointwise convergence of Cesaro means of two-variable functions with respect to unbounded Vilenkin systems","attachmentId":79223026,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924009/On_the_pointwise_convergence_of_Cesaro_means_of_two_variable_functions_with_respect_to_unbounded_Vilenkin_systems","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/68924009/On_the_pointwise_convergence_of_Cesaro_means_of_two_variable_functions_with_respect_to_unbounded_Vilenkin_systems"><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="68923985" 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/68923985/Almost_Everywhere_Convergence_of_Fej%C3%A9r_Means_of_L_1_Functions_on_Rarely_Unbounded_Vilenkin_Groups">Almost Everywhere Convergence of Fejér Means of L 1 Functions on Rarely Unbounded Vilenkin Groups</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, 2007</p><p class="ds-related-work--abstract ds2-5-body-sm">It is a highly celebrated problem in dyadic harmonic analysis the pointwise convergence of the Fejér (or (C, 1)) means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σ n f → f (n → ∞) for functions f ∈ L p , where p > 1</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 Fejér Means of L 1 Functions on Rarely Unbounded Vilenkin Groups","attachmentId":79223217,"attachmentType":"pdf","work_url":"https://www.academia.edu/68923985/Almost_Everywhere_Convergence_of_Fej%C3%A9r_Means_of_L_1_Functions_on_Rarely_Unbounded_Vilenkin_Groups","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/68923985/Almost_Everywhere_Convergence_of_Fej%C3%A9r_Means_of_L_1_Functions_on_Rarely_Unbounded_Vilenkin_Groups"><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="68923998" 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/68923998/Pointwise_convergence_of_the_Fej%C3%A9r_means_of_functions_on_unbounded_Vilenkin_groups">Pointwise convergence of the Fejér means of functions on unbounded Vilenkin groups</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">1999</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we prove that for a function f # L p (G m) where 1<p the Feje r means _ n f converge to f almost everywhere with respect to the character system of any (bounded or not) Vilenkin group G m .</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":"Pointwise convergence of the Fejér means of functions on unbounded Vilenkin groups","attachmentId":79223031,"attachmentType":"pdf","work_url":"https://www.academia.edu/68923998/Pointwise_convergence_of_the_Fej%C3%A9r_means_of_functions_on_unbounded_Vilenkin_groups","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/68923998/Pointwise_convergence_of_the_Fej%C3%A9r_means_of_functions_on_unbounded_Vilenkin_groups"><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="60326744" 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/60326744/Investigations_with_respect_to_the_maximal_operator_of_Fej%C3%A9r_means_on_Vilenkin_systems">Investigations with respect to the maximal operator of Fejér means on Vilenkin systems</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="142093004" href="https://independent.academia.edu/Istv%C3%A1nBlahota">István Blahota</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2006</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":"Investigations with respect to the maximal operator of Fejér means on Vilenkin systems","attachmentId":73822755,"attachmentType":"pdf","work_url":"https://www.academia.edu/60326744/Investigations_with_respect_to_the_maximal_operator_of_Fej%C3%A9r_means_on_Vilenkin_systems","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/60326744/Investigations_with_respect_to_the_maximal_operator_of_Fej%C3%A9r_means_on_Vilenkin_systems"><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="110444318" 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/110444318/Norm_Convergence_of_Double_Fej%C3%A9r_Means_on_Unbounded_Vilenkin_Groups">Norm Convergence of Double Fejér Means on Unbounded Vilenkin Groups</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, 2018</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we study the criterions of the uniform convergence and L-convergence of double Vilenkin-Fourier series. We also prove that these results are sharp.</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":"Norm Convergence of Double Fejér Means on Unbounded Vilenkin Groups","attachmentId":108260233,"attachmentType":"pdf","work_url":"https://www.academia.edu/110444318/Norm_Convergence_of_Double_Fej%C3%A9r_Means_on_Unbounded_Vilenkin_Groups","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/110444318/Norm_Convergence_of_Double_Fej%C3%A9r_Means_on_Unbounded_Vilenkin_Groups"><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="79992075" 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/79992075/On_the_H_L_typeness_of_the_maximal_functionof_Cesaro_means_of_two_parameter_integrable_functions_on_bounded_Vilenkin_groups">On the (H; L) typeness of the maximal functionof Cesaro means of two-parameter integrable functions on bounded Vilenkin groups</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="142093004" href="https://independent.academia.edu/Istv%C3%A1nBlahota">István Blahota</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Publicationes mathematicae</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we prove that the operator σ * f := sup n∈N 2 |σ n f | from the Hardy space H to L 1 (G m × G e m) is bounded, where the quotient of the coordinates of n is bounded. In other words σ * is of type (H, L).</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 (H; L) typeness of the maximal functionof Cesaro means of two-parameter integrable functions on bounded Vilenkin groups","attachmentId":86521831,"attachmentType":"pdf","work_url":"https://www.academia.edu/79992075/On_the_H_L_typeness_of_the_maximal_functionof_Cesaro_means_of_two_parameter_integrable_functions_on_bounded_Vilenkin_groups","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/79992075/On_the_H_L_typeness_of_the_maximal_functionof_Cesaro_means_of_two_parameter_integrable_functions_on_bounded_Vilenkin_groups"><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="68924028" 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/68924028/On_the_ae_convergence_of_Fourier_series_on_unbounded_Vilenkin_groups">On the ae convergence of Fourier series on unbounded Vilenkin groups</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">1999</p><p class="ds-related-work--abstract ds2-5-body-sm">It is well known that the 2 n th partial sums of the Walsh-Fourier series of an integrable function converges a.e. to the function. This result has been proved Sto by techniques known in the martingale theory. The author gave purely dyadic harmonic analysis" proof of this in the former volume of this journal G at. The Vilenkin groups are generalizations of the Walsh group. We prove the a.e. convergence S M n f ! f n ! 1; f 2 L 1 G m even in the case when G m is an unbounded Vilenkin group. The nowelty of this proof is that we use techniques, which are elementary in dyadic harmonic analysis. We do not use any technique in martingale theory used in the former proof Sto .</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 ae convergence of Fourier series on unbounded Vilenkin groups","attachmentId":79223028,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924028/On_the_ae_convergence_of_Fourier_series_on_unbounded_Vilenkin_groups","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/68924028/On_the_ae_convergence_of_Fourier_series_on_unbounded_Vilenkin_groups"><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="68924000" 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/68924000/Pointwise_convergence_of_double_Vilenkin_Fej%C3%A9r_means">Pointwise convergence of double Vilenkin-Fejér means</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">2000</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we prove that double Cesàro means of an integrable function on the product of bounded Vilenkin groups with respect to any double Vilenkin-like system converge to the function almost everywhere. The convergence is provided that the quotient of the indices is bounded.</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":"Pointwise convergence of double Vilenkin-Fejér means","attachmentId":79223188,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924000/Pointwise_convergence_of_double_Vilenkin_Fej%C3%A9r_means","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/68924000/Pointwise_convergence_of_double_Vilenkin_Fej%C3%A9r_means"><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="68924014" 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/68924014/Maximal_operators_of_Fej%C3%A9r_means_of_Vilenkin_Fourier_series">Maximal operators of Fejér means of Vilenkin-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">2006</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 operators of Fejér means of Vilenkin-Fourier series","attachmentId":79223025,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924014/Maximal_operators_of_Fej%C3%A9r_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-wsj-grid-card-view-pdf" href="https://www.academia.edu/68924014/Maximal_operators_of_Fej%C3%A9r_means_of_Vilenkin_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":79223054,"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":79223054,"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_79223054" 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="79990907" 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/79990907/Journal_of_Inequalities_in_Pure_and_Applied_Mathematics_MAXIMAL_OPERATORS_OF_FEJ%C3%89R_MEANS_OF_VILENKIN_FOURIER_SERIES">Journal of Inequalities in Pure and Applied Mathematics MAXIMAL OPERATORS OF FEJÉR 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="142093004" href="https://independent.academia.edu/Istv%C3%A1nBlahota">István Blahota</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2013</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":"Journal of Inequalities in Pure and Applied Mathematics MAXIMAL OPERATORS OF FEJÉR MEANS OF VILENKIN-FOURIER SERIES","attachmentId":86520973,"attachmentType":"pdf","work_url":"https://www.academia.edu/79990907/Journal_of_Inequalities_in_Pure_and_Applied_Mathematics_MAXIMAL_OPERATORS_OF_FEJ%C3%89R_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/79990907/Journal_of_Inequalities_in_Pure_and_Applied_Mathematics_MAXIMAL_OPERATORS_OF_FEJ%C3%89R_MEANS_OF_VILENKIN_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-related-work-sidebar-card" data-collection-position="1" data-entity-id="103541545" 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/103541545/Smoothness_conditions_and_integrability_theorems_on_bounded_Vilenkin_groups">Smoothness conditions and integrability theorems on bounded Vilenkin groups</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="40193071" href="https://murdoch.academia.edu/WalterBloom">Walter R Bloom</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of the Australian Mathematical Society, 1988</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":"Smoothness conditions and integrability theorems on bounded Vilenkin groups","attachmentId":103521505,"attachmentType":"pdf","work_url":"https://www.academia.edu/103541545/Smoothness_conditions_and_integrability_theorems_on_bounded_Vilenkin_groups","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/103541545/Smoothness_conditions_and_integrability_theorems_on_bounded_Vilenkin_groups"><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="2" data-entity-id="68924072" 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/68924072/Convergence_and_divergence_of_Fej%C3%A9r_means_of_Fourier_series_on_one_and_two_dimensional_Walsh_and_Vilenkin_groups">Convergence and divergence of Fejér means of Fourier series on one and two-dimensional Walsh and Vilenkin groups</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":"Convergence and divergence of Fejér means of Fourier series on one and two-dimensional Walsh and Vilenkin groups","attachmentId":79223192,"attachmentType":"pdf","work_url":"https://www.academia.edu/68924072/Convergence_and_divergence_of_Fej%C3%A9r_means_of_Fourier_series_on_one_and_two_dimensional_Walsh_and_Vilenkin_groups","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/68924072/Convergence_and_divergence_of_Fej%C3%A9r_means_of_Fourier_series_on_one_and_two_dimensional_Walsh_and_Vilenkin_groups"><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="3" data-entity-id="86377258" 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/86377258/a_e_Convergence_of_N%C3%B6rlund_means_with_respect_to_Vilenkin_systems_of_integrable_functions">a.e. Convergence of Nörlund means with respect to Vilenkin systems of integrable functions</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="127157369" href="https://independent.academia.edu/Tutberidze">Giorgi Tutberidze</a></div><p class="ds-related-work--metadata ds2-5-body-xs">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":"a.e. 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Convergence of N\"orlund means with respect to Vilenkin systems of integrable functions</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="195607967" href="https://independent.academia.edu/davitbaramidze">davit baramidze</a></div><p class="ds-related-work--metadata ds2-5-body-xs">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":"Lebesgue and Vilenkin-Lebesgue points and a. e. 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js-related-work-sidebar-card" data-collection-position="7" data-entity-id="110444334" 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/110444334/Norm_convergence_of_logarithmic_means_on_unbounded_Vilenkin_groups">Norm convergence of logarithmic means on unbounded Vilenkin groups</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">Banach Journal of Mathematical Analysis, 2018</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":"Norm convergence of logarithmic means on unbounded Vilenkin 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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 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">Miskolc Mathematical Notes, 2020</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 of the generalized Marcienkiwicz means with respect to two dimensional Vilenkin-like 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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/68924010/Maximal_operators_of_Fej%C3%A9r_means_of_double_Vilenkin_Fourier_series">Maximal operators of Fejér means of double 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">2007</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 operators of Fejér means of double Vilenkin--Fourier 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href="https://www.academia.edu/85415280/Summability_and_integrability_of_Vilenkin_series">Summability and integrability of Vilenkin 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="43632767" href="https://unsa-ba.academia.edu/Avdispahic">Muharem Avdispahić</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Collectanea Mathematica, 2000</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":"Summability and integrability of Vilenkin series","attachmentId":90120330,"attachmentType":"pdf","work_url":"https://www.academia.edu/85415280/Summability_and_integrability_of_Vilenkin_series","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download 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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":"On (C, 1) summability for Vilenkin-like systems","attachmentId":79223373,"attachmentType":"pdf","work_url":"https://www.academia.edu/68923999/On_C_1_summability_for_Vilenkin_like_systems","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/68923999/On_C_1_summability_for_Vilenkin_like_systems"><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="12" data-entity-id="68924002" 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/68924002/Pointwise_convergence_of_Fej%C3%A9r_means_on_compact_totally_disconnected_groups">Pointwise convergence of Fejér means on compact totally disconnected groups</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">1995</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":"Pointwise convergence of Fejér means on compact totally disconnected 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