Boas Type and Titchmarsh Type Theorems for Generalized Fourier-Bessel Transform

Abstract

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