Smoothing curve by orthogonal polynomials with non-differentiable functions as their weight functions

摘要

The interpolation by polynomial of degree n, p_n(x), may smooth any curve y = f (x) with the condition that this polynomial must passes through n +1 points of the curve y = f (x). This condition may prevent the curve of y = p_n(x) different from the exact curve of y = f (x) especially in the case that the points of f (x) are obtained from an experiment which may not be the exact point of the curve y = f (x). But the least square approximation by orthogonal polynomial q_n (x), may give the better shape than the interpolation by polynomial p_n(x). However the curve of y = q_n (x) may not pass through any exact point of the curve y = f (x). M.Podisuk, P.Rattanathanawan and P.Phataranavik, in [1], introduced the sequences of orthogonal polynomials with step functions as their weight functions but they did not use them for least square approximation. In this paper, the sequences of orthogonal polynomials with nondifferentiable functions as their weight functions will be used for least square approximation.