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Jackson’s inequalities in Mellin’s analysis

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Abstract

Let \( c\in {\mathbb {R}} \) and \( X_{c}^{2} \) be the set of functions \( f: {\mathbb {R}}_{+}\rightarrow {\mathbb {C}} \) such that \( f(\cdot )(\cdot )^{c-1/2} \) is square integrable in the Lebesgue’s sense over \( {\mathbb {R}}_{+} \). The Mellin integral transform of f is given by

$$\begin{aligned} {\mathcal {M}}[f](c+it):=\lim _{\rho \rightarrow +\infty }\int _{1/\rho }^{\rho }u^{c+it-1}f(u)du, \;\; t \in {\mathbb {R}}. \end{aligned}$$

The focus of this research is to prove analogs of Jackson’s direct and some inverse theorems in terms of best approximations of functions \( f \in X_{c}^{2} \) with bounded spectrum and the Mellin moduli of smoothness of all orders constructed by the Mellin Steklov operators.

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  1. Laboratory of Fundamental and Applied Mathematics Department of Mathematics, Faculty of Sciences Aïn Chock, University Hassan II, Casablanca, Morocco

    Othman Tyr & Radouan Daher

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  1. Othman Tyr
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Tyr, O., Daher, R. Jackson’s inequalities in Mellin’s analysis. Ann Univ Ferrara 70, 141–160 (2024). https://doi.org/10.1007/s11565-023-00462-9

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