On the estimation of functions belonging to Lipschitz class by block pulse functions and hybrid Legendre polynomials

Abstract

In this paper, block pulse functions and hybrid Legendre polynomials are introduced. The estimators of a function $f$ having first and second derivative belonging to $Lip_\alpha[a,b]$ class, $0 < \alpha \leq 1$, and $a$, $b$ are finite real numbers, by block pulse functions and hybrid Legendre polynomials have been calculated. These calculated estimators are new, sharp and best possible in wavelet analysis. An example has been given to explain the validity of approximation of functions by using the hybrid Legendre polynomials approximation method. A real-world problem of radioactive decay is solved using this hybrid Legendre polynomials approximation method. Moreover, the Hermite differential equation of order zero is solved by using hybrid Legendre polynomials approximation method to explain the importance and the application of the technique of this method.