<p>In the present paper Riesz sums of Fourier-Chebyshev rational integral operators with restrictions on the number of geometrically distinct poles are introduced. Approximation of the function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_73_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$(1-x)^\gamma$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>γ</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_73_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$\gamma \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, by this method is considered. Estimates of pointwise and uniform approximation are established,as well as asymptotic expressions for the uniform approximation majorant. Additionally, the optimal values of the parameters of the approximating function, at which the rate of decrease of the majorant is the greatest are found. In the case of Riesz sums of a polynomial Fourier-Chebyshev series, approximation of functions satisfying the Lipschitz condition of order <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_73_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\gamma$</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> on the segment <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_73_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$[-1,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> is investigated.</p>

On the Riesz summation of rational Fourier-Chebyshev integral operators and approximations of functions with a power singularity

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发表日期：

2025年4月22日

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P. Patseika,
Y. Rouba,
K. Smatrytski

摘要

In the present paper Riesz sums of Fourier-Chebyshev rational integral operators with restrictions on the number of geometrically distinct poles are introduced. Approximation of the function $(1-x)^\gamma$ ${(1 - x)}^{γ}$ , $\gamma \in (0,1)$ $γ \in (0, 1)$ , by this method is considered. Estimates of pointwise and uniform approximation are established,as well as asymptotic expressions for the uniform approximation majorant. Additionally, the optimal values of the parameters of the approximating function, at which the rate of decrease of the majorant is the greatest are found. In the case of Riesz sums of a polynomial Fourier-Chebyshev series, approximation of functions satisfying the Lipschitz condition of order $\gamma$ $γ$ on the segment $[-1,1]$ $[- 1, 1]$ is investigated.