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<p>The translation operator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1567_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(T^A\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mi>A</mi> </msup> </math></EquationSource> </InlineEquation> associated with the special affine Fourier transform (SAFT) <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1567_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathscr {F}_A\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> is introduced from harmonic analysis point of view. The analogues of Wendel’s theorem, Wiener theorem, Wiener-Tauberian theorem and Bernstein type inequality in the context of the SAFT are established. The shift invariant space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1567_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(V_A\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> associated with the special affine Fourier transform is introduced and studied along with sampling problems.</p>

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The system of translates and the special affine Fourier transform

  • 期刊论文

  • 发表日期：

    2025年4月25日

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  • Md Hasan Ali Biswas,
  • Frank Filbir,
  • Radha Ramakrishnan

摘要

The translation operator \(T^A\) T A associated with the special affine Fourier transform (SAFT) \(\mathscr {F}_A\) F A is introduced from harmonic analysis point of view. The analogues of Wendel’s theorem, Wiener theorem, Wiener-Tauberian theorem and Bernstein type inequality in the context of the SAFT are established. The shift invariant space \(V_A\) V A associated with the special affine Fourier transform is introduced and studied along with sampling problems.