Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\in \mathbb Z_+=\{0,1,\dots \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msub> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\nu >-1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>></mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. For a measurable on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb R_+$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </math></EquationSource> </InlineEquation> function f from the space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^1_{n,\nu }(\mathbb R_+)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>ν</mi> </mrow> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of special kind, we consider the generalized Fourier–Dunkl transform <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal F_{n,\nu ,d}(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>ν</mi> <mo>,</mo> <mi>d</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We prove a Boas type result about necessary and sufficient conditions for f to belong to the generalized uniform Lipschitz classes in terms of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal F_{n,\nu ,d}(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">F</mi> <mrow> <mi>n</mi> <mo>,</mo> <mi>ν</mi> <mo>,</mo> <mi>d</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Also, an analogue of classical Titchmarsh theorem describing Lipschitz classes in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> space in terms of Fourier transform is established in <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> space with power weight in the generalized Fourier–Dunkl transform setting.

Boas type and Titchmarsh type results for generalized Fourier–Dunkl transforms and symmetric Lipschitz classes

期刊论文
发表日期：

2026年1月30日

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Sergey Volosivets

摘要

Let $n\in \mathbb Z_+=\{0,1,\dots \}$ $n \in Z_{+} = {0, 1, \dots}$ , $\nu >-1/2$ $ν > - 1 / 2$ . For a measurable on $\mathbb R_+$ $R_{+}$ function f from the space $L^1_{n,\nu }(\mathbb R_+)$ $L_{n, ν}^{1} (R_{+})$ of special kind, we consider the generalized Fourier–Dunkl transform $\mathcal F_{n,\nu ,d}(f)$ $F_{n, ν, d} (f)$ . We prove a Boas type result about necessary and sufficient conditions for f to belong to the generalized uniform Lipschitz classes in terms of $\mathcal F_{n,\nu ,d}(f)$ $F_{n, ν, d} (f)$ . Also, an analogue of classical Titchmarsh theorem describing Lipschitz classes in $L^2$ $L^{2}$ space in terms of Fourier transform is established in $L^2$ $L^{2}$ space with power weight in the generalized Fourier–Dunkl transform setting.