Approximation of a Function f of Generalized Lipschitz Class by Its Extended Legendre Wavelet Series

Abstract

In this paper, two classes $${ Lip}_\alpha ^{(s)}[0, 1)$$ and $${ Lip}\xi [0, 1)$$ are introduced. These classes of functions are the generalization of the known Lipschitz class $${ Lip}_\alpha [0, 1), 0<\alpha \le 1$$ of functions. Four new estimators $$E_{\mu ^k,0}^{(\alpha )}(f)$$ , $$E_{\mu ^k,1}^{(\alpha )}(f)$$ , $$E_{\mu ^k,2}^{(\alpha )}(f)$$ and $$E_{\mu ^k,M}^{(\alpha )}(f)$$ of functions of $${ Lip}_\alpha ^{(s)}[0, 1)$$ class and $$E_{\mu ^k,0}^{(\xi )}(f),E_{\mu ^k,1}^{(\xi )}(f)$$ , $$E_{\mu ^k,2}^{(\xi )}(f)$$ and $$E_{\mu ^k,M}^{(\xi )}(f)$$ of functions of $${ Lip}\xi [0, 1)$$ class have been obtained. Five corollaries are deduced from the main theorems. These estimators are best possible in approximation of functions by wavelet methods.