On the Fourier–Dunkl Coefficients of Generalized Lipschitz Classes on the Interval $$[-1,1]$$

IF 1.1 3区数学 Q1 MATHEMATICS

Mediterranean Journal of Mathematics Pub Date : 2024-08-05 DOI:10.1007/s00009-024-02710-4

Othman Tyr

引用次数: 0

Abstract

In this paper, we consider $\mathcal {E}$ the set of all infinitely differentiable functions with compact support included on the interval $I=[-1,1]$. We use the distributions in $\mathcal {E}$, as a tool to prove the continuity of the Dunkl operator and the Dunkl translation. Some properties of the modulus of smoothness related to the Dunkl operator are verified. By means of generalized Dunkl–Lipschitz conditions on Dunkl–Sobolev spaces, a result of Younis on the torus, which is an analog of Titchmarsh’s theorem, is deduced as a special case. In addition, certain conditions and a characterization of the Dini–Lipschitz classes on I in terms of the behavior of their Fourier–Dunkl coefficients are derived.

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论区间 $$[-1,1]$$ 上广义 Lipschitz 类的傅立叶-邓克尔系数

在本文中，我们认为 $\mathcal {E}$ 是包含在区间 $I=[-1,1]$上的所有具有紧凑支持的无穷微分函数的集合。我们用 $\mathcal {E}$ 中的分布作为工具来证明邓克尔算子和邓克尔平移的连续性。我们还验证了与邓克尔算子相关的平滑模量的一些性质。通过 Dunkl-Sobolev 空间上的广义 Dunkl-Lipschitz 条件，作为特例推导出了 Younis 在环上的一个结果，它是 Titchmarsh 定理的类似物。此外，还根据傅里叶-敦克尔系数的行为推导出了 I 上 Dini-Lipschitz 类的某些条件和特征。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mediterranean Journal of Mathematics 数学-数学

CiteScore

1.80

自引率

0.00%

发文量

261

审稿时长

6-12 weeks

期刊介绍： The Mediterranean Journal of Mathematics (MedJM) is a publication issued by the Department of Mathematics of the University of Bari. The new journal replaces the Conferenze del Seminario di Matematica dell’Università di Bari which has been in publication from 1954 until 2003. The Mediterranean Journal of Mathematics aims to publish original and high-quality peer-reviewed papers containing significant results across all fields of mathematics. The submitted papers should be of medium length (not to exceed 20 printed pages), well-written and appealing to a broad mathematical audience. In particular, the Mediterranean Journal of Mathematics intends to offer mathematicians from the Mediterranean countries a particular opportunity to circulate the results of their researches in a common journal. Through such a new cultural and scientific stimulus the journal aims to contribute to further integration amongst Mediterranean universities, though it is open to contribution from mathematicians across the world.