Abstract <p> Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5124_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="108" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha\in (-1/2,\infty )\)</EquationSource> </InlineEquation> and let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5124_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\chi _r\)</EquationSource> </InlineEquation> denote the indicator function of the segment<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5124_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\([-r,r]\)</EquationSource> </InlineEquation>. We obtain new two-radii theorems for the Besselconvolution operator <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5124_Article_IEq4.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\rightarrow f\overset {\alpha }\star \chi _r\)</EquationSource> </InlineEquation> that are related to quasi-analytic classes offunctions. We establish a local analog of the two-radii theorem for functions <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5124_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\)</EquationSource> </InlineEquation> that satisfy the convolution inequalities<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5124_Article_IEq6.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\overset{\alpha }\star \chi _{r_1}\geq 0\)</EquationSource> </InlineEquation> and<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12002_2025_5124_Article_IEq7.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\overset{\alpha }\star \chi _{r_2}\leq 0\)</EquationSource> </InlineEquation>. We alsopresent applications of these results to uniqueness theorems for solutions of the Cauchy problemfor the generalized Euler–Poisson–Darboux equation and to closure theorems for generalizedshifts.</p>

Averages Relative to the Bessel Convolution and Their Applications

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发表日期：

2025年6月1日

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G. V. Krasnoschekikh,
Vit. V. Volchkov

摘要

Abstract

Let \(\alpha\in (-1/2,\infty )\) and let \(\chi _r\) denote the indicator function of the segment \([-r,r]\) . We obtain new two-radii theorems for the Besselconvolution operator \(f\rightarrow f\overset {\alpha }\star \chi _r\) that are related to quasi-analytic classes offunctions. We establish a local analog of the two-radii theorem for functions \(f\) that satisfy the convolution inequalities \(f\overset{\alpha }\star \chi _{r_1}\geq 0\) and \(f\overset{\alpha }\star \chi _{r_2}\leq 0\) . We alsopresent applications of these results to uniqueness theorems for solutions of the Cauchy problemfor the generalized Euler–Poisson–Darboux equation and to closure theorems for generalizedshifts.