APPROXIMATIONS IN HÖLDER’S CLASS AND SOLUTION OF BESSEL’S DIFFERENTIAL EQUATIONS BY EXTENDED HAAR WAVELET

Abstract

In this paper, extended Haar wavelet has been introduced in the interval [0, λ), λ > 0. It reduces to classical Haar wavelet for λ = 1. The orthonormality of extended Haar wavelets has been discussed. The convergence analysis of an extended Haar wavelet series of a function f belonging to Hölder’s classes Hα [0, λ) & H2α [0, λ) have been studied. Consequently, the approximations of function f belonging to the generalised Hölder’s class have been estimated. The solutions of Bessel’s differential equation of order zero have been obtained by the extended Haar operational matrix method for λ = 1 & 2. These solutions for λ = 1 & 2 are compared with their exact solutions. It is observed that the extended Haar wavelet solutions and their exact solutions are almost the same. This validates the adopted procedure for solutions of Bessel’s differential equation by an extended Haar operational matrix. This is a significant achievement in wavelet analysis. © Poincare Publishers.