On the equivalence of K-functionals and modulus of smoothness constructed by the generalized Fourier–Bessel transform

Using a generalized translation operator, we define generalized modulus of smoothness in the space Lα,n2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {L}}_{\alpha , n}^{2}$$\end{document}. Based on the generalized Bessel operator we define Sobolev-type space and K-functionals. The main result of this paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness for the generalized Fourier–Bessel transform.