<p>We prove direct and inverse approximation theorems for modulus of smoothness of order <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1015_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$m>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> based on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1015_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$q$</EquationSource> <EquationSource Format="MATHML"><math> <mi>q</mi> </math></EquationSource> </InlineEquation>-Bessel translation and best approximation by functions with <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1015_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$q$</EquationSource> <EquationSource Format="MATHML"><math> <mi>q</mi> </math></EquationSource> </InlineEquation>-Bessel Fourier transform vanishing outside <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1015_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$[0,\sigma ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1015_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> space on <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1015_Article_IEq8.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^+_q$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>q</mi> <mo>+</mo> </msubsup> </math></EquationSource> </InlineEquation> with power weight <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1015_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$x^{2\nu +1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mrow> <mn>2</mn> <mi>ν</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2024_1015_Article_IEq10.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Some application of obtained results are given.</p>

Direct and inverse approximation theorems connected with the q-Bessel Fourier transform in weighted $L^2$ space

期刊论文
发表日期：

2025年1月28日

查看全文

Sergey Volosivets,
Yulia Krotova

摘要

We prove direct and inverse approximation theorems for modulus of smoothness of order $m>0$ $m > 0$ based on $q$ $q$ -Bessel translation and best approximation by functions with $q$ $q$ -Bessel Fourier transform vanishing outside $[0,\sigma ]$ $[0, σ]$ in $L^2$ $L^{2}$ space on $\mathbb {R}^+_q$ $R_{q}^{+}$ with power weight $x^{2\nu +1}$ $x^{2 ν + 1}$ , $\nu \ge 0$ $ν \geq 0$ . Some application of obtained results are given.

Direct and inverse approximation theorems connected with the q-Bessel Fourier transform in weighted \(L^2\) space

摘要