Orthogonal Functions Based on Chebyshev Polynomials

Abstract

It is known that Chebyshev polynomials are an orthogonal set associated with a certain weight function. In this paper, we present an approach for the contruction of a special wavelet function as well as a special scaling function. Main tool of the special wavelet is a first kind Chebyshev polynomial. Based on Chebyshev polynomials and their zero, we define our scaling function and wavelets, and by using Christoffel-Darboux formula for Chebyshev polynomials, we prove that these functions are orthogonal. Finally, we provide several examples of scaling function and wavelets for illustration. Keywords: Chebyshev polynomial; Christoffel-Darboux formula; Wavelets; and Scaling function. 2010 Mathematics Subject Classification 76D04.