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In this paper, several direct and inverse theorems in terms of the best approximations of functions and moduli of smoothness are proved concerning the approximation of functions from the space [Formula omitted] by partial sums of Jacobi-Dunkl series. For this purpose, we use the generalized Jacobi-Dunkl translation operator which was defined by Vinogradov., Author(s): Othman Tyr [sup.1], Radouan Daher [sup.1] Author Affiliations: (1) https://ror.org/001q4kn48, grid.412148.a, 0000 0001 2180 2473, Department of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, , Casablanca, Morocco [...]

This paper uses some basic concepts and results on harmonic analysis associated with Heckman-Opdam theory on [Formula omitted] to study some types of wavelet packets and the corresponding wavelet transforms. More precisely, we study two types of wavelet packets and the corresponding wavelet transforms related to the hypergeometric Fourier transform on W-invariant functions, and we prove the Plancherel, Calderón, and the reconstruction formulae for these transforms., Author(s): Othman Tyr [sup.1], Radouan Daher [sup.1], Abdelaali Dades [sup.1], Yassine Fantasse [sup.1] Author Affiliations: (1) https://ror.org/001q4kn48, grid.412148.a, 0000 0001 2180 2473, Department of Mathematics, Faculty of Sciences Aïn Chock, [...]

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]

This paper uses some basic concepts and results on harmonic analysis associated with Heckman-Opdam theory on R d to study some types of wavelet packets and the corresponding wavelet transforms. More precisely, we study two types of wavelet packets and the corresponding wavelet transforms related to the hypergeometric Fourier transform on W-invariant functions, and we prove the Plancherel, Calderón, and the reconstruction formulae for these transforms. [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]