Jackson’s inequalities in Mellin’s analysis

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.