Search

This work is devoted to the study of the integrability of Mellin transforms under some sufficient and necessary conditions in the space $ X_{c}^{p} $ X c p of Lebesgue measurable functions f on $ \mathbb {R}_+=(0,+\infty) $ R + = (0 , + ∞) such that \[ \int_{0}^{+\infty}|u^{c}f(u)|^{p}\frac{{\rm d}u}{u} ∫ 0 + ∞ | u c f (u) | p d u u < ∞ , c ∈ R (1 < p ≤ 2). These results are inspired by a Móricz-type result in classical Fourier analysis, which generalizes a well-known theorem of Titchmarsh. [ABSTRACT FROM AUTHOR]

The problem of weighted integrability of the Fourier–Laguerre transform in terms of the moduli of smoothness related to generalized translations is considered. Sufficient conditions are given to solve this problem. These results generalize a famous Titchmarsh's theorem and Younis' theorem, due to Negzaoui in the Laguerre hypergroup. Also some results connected with the integrability of Fourier–Laguerre transforms of Laguerre convolutions are given. [ABSTRACT FROM AUTHOR]

The purpose of this work is to prove analogues of the classical Titchmarsh theorem and Younis' theorem associated with the Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition of a certain order in suitable weighted spaces L p ([ 0 , + ∞)) , 1 < p ≤ 2. For this purpose, we use a generalized translation operator defined by Flensted-Jensen and Koornwinder. [ABSTRACT FROM AUTHOR]