Wavelet Approximation of Functions Belonging to Generalized Lipschitz Class by Extended Haar Wavelet Series and its Application in the Solution of Bessel Differential Equation Using Haar Operational Matrix

Abstract

In this paper, a class $$Lip_\alpha ^{(s)}[0,1]$$ is introduced. This class of functions is a generalization of known Lipschitz class, i.e., $$Lip_\alpha [0,1], 0<\alpha \le 1$$ of functions. Four new estimators $$E_N^{(1)}(f), E_{J,N}^{(2)}(f), E_N^{(3)}(f)$$ and $$E_N^{(4)}(f)$$ of functions of $$Lip_\alpha [0,1]$$ and $$Lip_\alpha ^{(s)}[0,1]$$ classes have been obtained. These estimators are sharper and best possible in the approximation of functions by wavelet methods. The developed estimators, the solution of Bessel differential equation of order zero by Haar wavelet operational matrix and its comparison with the exact solution are the main significant achievements of this research paper.