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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]

In this paper, we consider ξ the set of infinitely differentiable 2π -periodic functions on the circle T = R/2πZ. We use the distributions in ξ, as a tool to prove the continuity of the Jacobi-Dunkl operator. We obtain a generalization of the classical Titchmarsh theorem for the Jacobi-Dunkl coefficients of a set of functions satisfying Lipschitz conditions, with the use of the generalized Jacobi-Dunkl translation operator defined by Vinogradov. In addition, we introduce the discrete Jacobi-Dunkl Dini-Lipschitz class and we obtain an analogue of Younis’ theorem in this occurrence. [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]

In this paper, using the q -Dunkl harmonic analysis introduced in Bettaibi and Bettaieb [q-Analog of the Dunkl transform on the real line, Tamsui Oxf. J. Math. Sci. 25(2) (2007) 117–205] and motivated by Wong's approach [M. W. Wong, Wavelet Transforms and Localization Operators, Vol. 136 (Springer Science & Business Media, Berlin, 2002)], we first define and study the q -wavelets, the continuous q -wavelet transforms. Next, we introduce the two-wavelet localization operators, the Schatten–von Neumann properties of these localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented. [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]

In the present paper, we obtain some new results, and we generalize some known results for the Hausdorff operators. We have studied the generalized Hausdorff operators H α , φ on the Dunkl-type homogeneous weighted Herz spaces K ̇ α , q β , p ℝ and Dunkl Herz-type Hardy spaces H K ̇ α , q β , p , N ℝ . We have determined simple sufficient conditions for these operators to be bounded on these spaces. As applications, we provide necessary and sufficient conditions for generalized Cesàro operator to be bounded on K ̇ α , q β , p ℝ and Hardy inequality for K ̇ α , q β , p ℝ . [ABSTRACT FROM AUTHOR]