The main objective of this work is to investigate among other things the validity of the classical Titchmarsh’s theorem (<CitationRef CitationID="CR19">1937</CitationRef>, Theorem 84) and Younis’ theorem (Internat. J. Math. Math. Sci. 9(2), 301–312, <CitationRef CitationID="CR22">1986</CitationRef>, Theorem 3.3) on the unit sphere <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7576_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\varSigma _{m-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Σ</mi> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7576_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. More precisely, we prove those theorems on the image under the discrete Fourier-Laplace transform of a set of functions satisfying a generalized Lipschitz and Dini-Lipschitz condition in the space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7576_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$L_{p}(\varSigma _{m-1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>Σ</mi> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7576_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$ 1 \le p \le 2 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. For this purpose, we use a generalized spherical shift defined by Rudin (Trans. Amer. Math. Soc. 68, 287–303, <CitationRef CitationID="CR18">1950</CitationRef>).

ON THE SPHERICAL TRANSFORMATION OF FUNCTIONS FROM LIPSCHITZ AND DINI-LIPSCHITZ CLASSES

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发表日期：

2025年2月6日

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Othman Tyr,
Radouan Daher

摘要

The main objective of this work is to investigate among other things the validity of the classical Titchmarsh’s theorem (1937, Theorem 84) and Younis’ theorem (Internat. J. Math. Math. Sci. 9(2), 301–312, 1986, Theorem 3.3) on the unit sphere $\varSigma _{m-1}$ $Σ_{m - 1}$ , $m\ge 3$ $m \geq 3$ . More precisely, we prove those theorems on the image under the discrete Fourier-Laplace transform of a set of functions satisfying a generalized Lipschitz and Dini-Lipschitz condition in the space $L_{p}(\varSigma _{m-1})$ $L_{p} (Σ_{m - 1})$ , $ 1 \le p \le 2 $ $1 \leq p \leq 2$ . For this purpose, we use a generalized spherical shift defined by Rudin (Trans. Amer. Math. Soc. 68, 287–303, 1950).