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<p>This research investigates the two-sided quaternionic Dunkl transform of functions that satisfy specific Lipschitz conditions. In particular, we focus on Lipschitz functions belonging to the function space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{2}\left( \mathbb {R}^{2}, \mathbb {H}\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mfenced close=")" open="("> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo>,</mo> <mi mathvariant="double-struck">H</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and study how these smoothness constraints influence the behavior of their quaternionic Dunkl transforms. Motivated by Theorems 84 and 85 of Titchmarsh, which characterize the transforms of Lipschitz functions on the real line, we extend these classical results to the quaternionic Dunkl framework, providing a natural generalization in the context of quaternion-valued functions and Dunkl-type analysis.</p>

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Titchmarsh Theorem for the Two-Sided Quaternionic Dunkl Transform

  • 期刊论文

  • 发表日期：

    2025年12月2日

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  • M. Younus Bhat,
  • A. Achak,
  • N. Safouane

摘要

This research investigates the two-sided quaternionic Dunkl transform of functions that satisfy specific Lipschitz conditions. In particular, we focus on Lipschitz functions belonging to the function space \(L^{2}\left( \mathbb {R}^{2}, \mathbb {H}\right) \) L 2 R 2 , H and study how these smoothness constraints influence the behavior of their quaternionic Dunkl transforms. Motivated by Theorems 84 and 85 of Titchmarsh, which characterize the transforms of Lipschitz functions on the real line, we extend these classical results to the quaternionic Dunkl framework, providing a natural generalization in the context of quaternion-valued functions and Dunkl-type analysis.