Boas-type theorems for the free metaplectic transform

Abstract

In this study, we focus on the free metaplectic transform and its implications on the properties of functions. The free metaplectic transform is a generalization of the Fourier transform that allows us to analyze the behavior of functions in the metaplectic domain. F. Moricz previously investigated the properties of functions \(f\in L^1({\mathbb {R}})\) whose Fourier transforms \(\widehat{f}\) belong to \(L^1({\mathbb {R}})\). He established certain sufficient conditions based on \(\widehat{f}\) to determine whether f belongs to the Lipschitz classes \({\text {Lip}}(\gamma )\) and \({\text {lip}}(\gamma )\), where \(0 < \gamma \le 1\), or the Zygmund classes \({\text {Zyg}}(\gamma )\) and \({\text {zyg}}(\gamma )\), where \(0 < \gamma \le 2\). In this study, our aim is to extend these findings and explore the properties of functions in relation to the free metaplectic transform.