Approximation of signals (functions) belonging to certain Lipschitz classes by almost Riesz means of its Fourier series

To start with, signals are dealt with as functions of one variable and images are shown by elements of two variables. The investigation of these ideas is directly related to the transpiring area of information technology. The approximation properties of the periodic signals in Lr(r≥1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{r}\ (r \geq1)$\end{document}-spaces, Lipschitz classes Lipα, Lip(α,r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{Lip}(\alpha, r)$\end{document}, Lip(ξ(t),r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{Lip}(\xi(t), r)$\end{document}, and a weighted Lipschitz class W(Lr,ξ(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$W(L^{r}, \xi( t))$\end{document} through a Fourier series, known as the Fourier approximation in the approximation theory, have wide applications in digital filters and signal analysis. The goal of our paper is to concentrate on the approximation properties of the periodic signals (functions) in the Lipschitz classes by almost Riesz means of the Fourier series associated with the function f. We additionally take note of the fact that our outcomes give sharper estimates than the estimates in some of the known results.