<p>Let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\varphi_n\}^\infty_{n=1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>φ</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> be a real-valued orthonormal system in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$L^2[0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="74" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\alpha\leq 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq5.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="75" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\varepsilon<\alpha$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>ε</mi> <mo><</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$\{c_n(f)\}^\infty_{n=1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>c</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> be a sequence of Fourier coefficients of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in L^2[0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with respect to <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\varphi_n\}^\infty_{n=1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>φ</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation>. We prove a sufficient condition on <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\varphi_n\}^\infty_{n=1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>φ</mi> <mi>n</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> such that the series <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq10.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="128" /> </InlineMediaObject> <EquationSource Format="TEX">$\sum^\infty_{k=1}k^{2(\alpha-\varepsilon)}c^2_k(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <msup> <mi>k</mi> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>α</mi> <mo>-</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </msup> <msubsup> <mi>c</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> converges for any <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10476_2025_91_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in Lip (\alpha)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi>L</mi> <mi>i</mi> <mi>p</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We check that the trigonometric system and the Haar system satisfy this condition. On the other hand, the condition is not fulfilled in general.</p>

Fourier coefficients of functions with respect to general othonormal systems from classes $ Lip (\alpha)$

期刊论文
发表日期：

2025年6月12日

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B. Golubov,
S. Volosivets

摘要

Let $\{\varphi_n\}^\infty_{n=1}$ ${φ_{n}}_{n = 1}^{\infty}$ be a real-valued orthonormal system in $L^2[0,1]$ $L^{2} [0, 1]$ , $0<\alpha\leq 1$ $0 < α \leq 1$ , $0<\varepsilon<\alpha$ $0 < ε < α$ , and let $\{c_n(f)\}^\infty_{n=1}$ ${c_{n} (f)}_{n = 1}^{\infty}$ be a sequence of Fourier coefficients of $f\in L^2[0,1]$ $f \in L^{2} [0, 1]$ with respect to $\{\varphi_n\}^\infty_{n=1}$ ${φ_{n}}_{n = 1}^{\infty}$ . We prove a sufficient condition on $\{\varphi_n\}^\infty_{n=1}$ ${φ_{n}}_{n = 1}^{\infty}$ such that the series $\sum^\infty_{k=1}k^{2(\alpha-\varepsilon)}c^2_k(f)$ $\sum_{k = 1}^{\infty} k^{2 (α - ε)} c_{k}^{2} (f)$ converges for any $f\in Lip (\alpha)$ $f \in L i p (α)$ . We check that the trigonometric system and the Haar system satisfy this condition. On the other hand, the condition is not fulfilled in general.

Fourier coefficients of functions with respect to general othonormal systems from classes \( Lip (\alpha)\)

摘要