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data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="90714653" data-landing_url="https://www.academia.edu/86207549/On_the_uniform_convergence_of_Fourier_transforms_on_groups" 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="16400216" 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/16400216/On_Twisted_Fourier_Analysis_and_Convergence_of_Fourier_Series_on_Discrete_Groups">On Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups</a><div class="ds-related-work--metadata"><a 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For amenable groups, Følner nets give the key to Fejér summation. We show that Abel-Poisson summation holds for a large class of groups, including e.g. all Coxeter groups and all Gromov hyperbolic groups. As a tool in our presentation, we introduce notions of polynomial and subexponential H-growth for countable groups w.r.t. proper scale functions, usually chosen as length functions. These coincide with the classical notions of growth in the case of amenable groups.</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 Twisted Fourier Analysis and Convergence of Fourier Series on Discrete Groups","attachmentId":42500779,"attachmentType":"pdf","work_url":"https://www.academia.edu/16400216/On_Twisted_Fourier_Analysis_and_Convergence_of_Fourier_Series_on_Discrete_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/16400216/On_Twisted_Fourier_Analysis_and_Convergence_of_Fourier_Series_on_Discrete_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="1" data-entity-id="61752013" 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/61752013/A_Paley_Wiener_theorem_for_real_reductive_groups">A Paley-Wiener theorem for real reductive 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="107515698" href="https://independent.academia.edu/ArthurJamesMichael">James Michael Arthur</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Acta Mathematica, 1983</p><p class="ds-related-work--abstract ds2-5-body-sm">where 6n is the modular function of B and chin(s, A) is Harish-Chandra's c function. We will extend Enln.,(., ~P, A) to a r spherical function on G_ = KAo(B) K, A PALEY-WIENER THEOREM FOR REAL REDUCTIVE GROUPS an open dense subset of G. It will then turn out that for FEPW(G, r) and xEG_, EBIn, ~(x,/aB(A) F(A), A) = EBIn, l(x, :tB(sA) F(sA), sA). As a function of A this expression will be meromorphic with poles along hyperplanes of the form (13, A)=r, for/~ a root of (G, A0) and r E R (Corollary 1.6.3). We shall also show that only finitely many of these poles intersect the negative chamber,-ao*(B), in (Lemma 1.5.3). For this introduction, let us assume that none of the poles meet the imaginary space ia~. Then for x E G_, F~n)(x) = fa~ Esln" l(x' F~B(A) F(A), A) dA. Let X be a point in general position in the chamber-a~(B) which is far from any of the walls. In Theorem II. I. 1 we will show that the function F~(x) = l EsIB, I(x,I~n(A)F(A),A)dA, xEG_, JX+ia~ is supported on a subset of G_ whose closure in G is compact. It will be our candidate for f(x). The difference F"(x)-F~n}(x) will be a sum of residues. Each one will be an integral over an affine space Xr+ib, where X r E a~ and b is a linear subspace of ct~' of dimension n-1. (T will belong to some indexing set.) We will group the residual integrals into sums corresponding to the W0orbits among the spaces b. Now for any W0-orbit of spaces b there is an associated class ~ of parabolic subgroups of G, of parabolic rank n-1. We might expect that the corresponding sum of residual integrals should equal f~(x). However, to have any hope of this we will need to replace each Xr by a vector which is orthogonal to b. Let Ar be the vector in Xr+ 19 which is orthogonal to b. Let F~,(x) be the sum of all the residual integrals, taken now over the contour Ar+ib, for which b belongs to the Wo-orbit associated to ~. (Again, we make the simplifying assumption that each residual integrand is regular on AT+ib.</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 Paley-Wiener theorem for real reductive groups","attachmentId":74709227,"attachmentType":"pdf","work_url":"https://www.academia.edu/61752013/A_Paley_Wiener_theorem_for_real_reductive_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/61752013/A_Paley_Wiener_theorem_for_real_reductive_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="2" data-entity-id="96033039" 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/96033039/Exponential_estimates_for_the_conjugate_function_on_locally_compact_abelian_groups">Exponential estimates for the conjugate function on locally compact abelian 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="44110821" href="https://independent.academia.edu/NakhleAsmar">Nakhlé Asmar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Canadian Mathematical Bulletin, 1990</p><p class="ds-related-work--abstract ds2-5-body-sm">Let G be a locally compact Abelian group, with character group X. Suppose that X contains a measurable order P. For the conjugate function of f is the function whose Fourier transform satisfies the identity for almost all χ in X where sgnp(χ) = - 1 , 0, 1, according as We prove that, when f is bounded with compact support, the conjugate function satisfies some weak type inequalities similar to those of the Hilbert transform of a bounded function with compact support in ℝ. As a consequence of these inequalities, we prove that possesses strong integrability properties, whenever f is bounded and G is compact. 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groups","attachmentId":106608310,"attachmentType":"pdf","work_url":"https://www.academia.edu/108147261/Absolute_convergence_of_Fourier_series_on_totally_disconnected_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/108147261/Absolute_convergence_of_Fourier_series_on_totally_disconnected_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="5" data-entity-id="58405037" 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/58405037/Functions_that_operate_in_the_Fourier_algebra_of_a_compact_group">Functions that operate in the Fourier algebra of a compact group</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="39545224" href="https://independent.academia.edu/CharlesDunkl">Charles Dunkl</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Proceedings of the American Mathematical Society, 1969</p><p class="ds-related-work--abstract ds2-5-body-sm">Only real-analytic functions operate in the Fourier algebra of any compact group that has an infinite abelian subgroup. This extends the theorems of Helson, Kahane, Katznelson, and Rudin [4] which apply to the algebra of absolutely convergent Fourier series on compact abelian groups. The Fourier algebra of a locally compact group has been studied by H. Mirkil [ó], W. F. Stinespring [9], R. A. Mayer [5], C. Herz [3], and most thoroughly by P. Eymard [l]. We will state here the relevant definitions and facts, and prove that the restriction of the Fourier algebra to a closed subgroup is the Fourier algebra of the subgroup, and use this to lift up the theorem on operating functions.</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":"Functions that operate in the Fourier algebra of a compact group","attachmentId":72838146,"attachmentType":"pdf","work_url":"https://www.academia.edu/58405037/Functions_that_operate_in_the_Fourier_algebra_of_a_compact_group","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/58405037/Functions_that_operate_in_the_Fourier_algebra_of_a_compact_group"><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="109451115" 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/109451115/Fundamental_Theorems_of_Fourier_Stieltjes_Transform_Defined_by_Induced_Representation_on_Locally_Compact_Group">Fundamental Theorems of Fourier-Stieltjes Transform Defined by Induced Representation on Locally Compact Group</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="137244740" href="https://independent.academia.edu/Hounkonnou">Norbert Hounkonnou</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2022</p><p class="ds-related-work--abstract ds2-5-body-sm">This work addresses an extension of Fourier-Stieltjes transform of a vector measure defined on compact groups to locally compact groups, by using a group representation induced by a representation of one of its compact subgroups.</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":"Fundamental Theorems of Fourier-Stieltjes Transform Defined by Induced Representation on Locally Compact Group","attachmentId":107572730,"attachmentType":"pdf","work_url":"https://www.academia.edu/109451115/Fundamental_Theorems_of_Fourier_Stieltjes_Transform_Defined_by_Induced_Representation_on_Locally_Compact_Group","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/109451115/Fundamental_Theorems_of_Fourier_Stieltjes_Transform_Defined_by_Induced_Representation_on_Locally_Compact_Group"><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="126166707" 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/126166707/On_Sj%C3%B6lin_Soria_Antonov_type_extrapolation_for_locally_compact_groups_and_a_e_convergence_of_Vilenkin_Fourier_series">On Sjölin–Soria–Antonov type extrapolation for locally compact groups and a.e. convergence 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="298744359" href="https://independent.academia.edu/GiorgiOniani7">Giorgi Oniani</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Acta Mathematica Hungarica</p><p class="ds-related-work--abstract ds2-5-body-sm">Sjölin-Soria-Antonov type extrapolation theorem for locally compact σ-compact non-discrete groups is proved. As an application of this result it is shown that the Fourier series with respect to the Vilenkin orthonormal systems on the Vilenkin groups of bounded type converge almost everywhere for functions from the class L log + L log + log + log + L. Let (X, µ) be a measure space. Denote by: • L 0 (X, µ) the class of all measurable functions f : X → [−∞, ∞]; • Φ the set of all increasing continuous functions ϕ : [0, ∞) → [0, ∞) with ϕ(0) = 0 and lim inf u→∞ ϕ(u)/u > 0; • ϕ(L)(X, µ) the class of all measurable functions f : X → [−∞, ∞] for which X ϕ(|f |)dµ < ∞; • χ E the characteristic function of a set E ⊂ X.</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 Sjölin–Soria–Antonov type extrapolation for locally compact groups and a.e. convergence of Vilenkin–Fourier series","attachmentId":120085344,"attachmentType":"pdf","work_url":"https://www.academia.edu/126166707/On_Sj%C3%B6lin_Soria_Antonov_type_extrapolation_for_locally_compact_groups_and_a_e_convergence_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/126166707/On_Sj%C3%B6lin_Soria_Antonov_type_extrapolation_for_locally_compact_groups_and_a_e_convergence_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-wsj-grid-card" data-collection-position="8" data-entity-id="51635678" 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/51635678/Motion_Groups_and_Absolutely_Convergent_Fourier_Transforms">Motion Groups and Absolutely Convergent Fourier Transforms</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="42712098" href="https://independent.academia.edu/RitaPini">Rita Pini</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of the Australian Mathematical Society, 1987</p><p class="ds-related-work--abstract ds2-5-body-sm">According to an extension of a classical theorem of Bernstein, due to C. Herz, a function on R" belonging to a Besov space of appropriate order has an absolutely convergent Fourier transform. We establish extensions of this result to Cartan motion groups, for Besov spaces defined with respect to both isotropic and non-isotropic differences.</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":"Motion Groups and Absolutely Convergent Fourier Transforms","attachmentId":69273353,"attachmentType":"pdf","work_url":"https://www.academia.edu/51635678/Motion_Groups_and_Absolutely_Convergent_Fourier_Transforms","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/51635678/Motion_Groups_and_Absolutely_Convergent_Fourier_Transforms"><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="49544872" 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/49544872/Multipliers_on_spaces_of_functions_on_compact_groups_with_p_summable_Fourier_transforms">Multipliers on spaces of functions on compact groups with p-summable Fourier transforms</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="152333134" href="https://independent.academia.edu/sanjivgupta21">sanjiv gupta</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Bulletin of the Australian Mathematical Society, 1993</p><p class="ds-related-work--abstract ds2-5-body-sm">Let G be a compact abelian group with dual group Γ. For 1 ≤ p &lt; ∞, denote by Ap(G) the space of integrable functions on G whose Fourier transforms belong to lp(Γ). We investigate several problems related to multipliers from Ap(G) to Aq(G). In particular, we prove that (Ap, Ap) ⊊ (Aq, Aq). For the circle group, we characterise permutation invariant multipliers from Ap to Ar for 1 ≤ r ≤ 2.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Multipliers on spaces of functions on compact groups with p-summable Fourier transforms","attachmentId":67873285,"attachmentType":"pdf","work_url":"https://www.academia.edu/49544872/Multipliers_on_spaces_of_functions_on_compact_groups_with_p_summable_Fourier_transforms","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/49544872/Multipliers_on_spaces_of_functions_on_compact_groups_with_p_summable_Fourier_transforms"><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":90714653,"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":90714653,"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_90714653" 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. 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II</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="63569879" href="https://independent.academia.edu/RichardDMosak">Richard D . Mosak</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1973</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":"Duality and harmonic analysis on central topological groups. II","attachmentId":107704918,"attachmentType":"pdf","work_url":"https://www.academia.edu/109640985/Duality_and_harmonic_analysis_on_central_topological_groups_II","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/109640985/Duality_and_harmonic_analysis_on_central_topological_groups_II"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="6" data-entity-id="22654930" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" 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class="ds-related-work--metadata ds2-5-body-xs">Open Journal of Mathematical Sciences</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 vector Fourier multipliers for compact groups","attachmentId":70854259,"attachmentType":"pdf","work_url":"https://www.academia.edu/54535373/On_the_vector_Fourier_multipliers_for_compact_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/54535373/On_the_vector_Fourier_multipliers_for_compact_groups"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" 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