A class of discrete orthogonal piecewise polynomials and its applications

For computer geometric figures representation,there is Gibbs phenomenon if continuous basis functions are used to approximate the discontinuous signals with breakpoints.The rate of convergence is very slow if Walsh basis functions are used to represent the discontinuous signals.Thus a class of discrete piecewise orthogonal polynomials basis(DPTB) was constructed from discrete orthogonal Tchebichef polynomials,whose breakpoints appear at(N-1)/2p.Since this class of basis consists of smooth and piecewise polynomials parts,finite discrete geometric figures with breakpoints at(N-1)/2p can be precisely expressed by using the constructed orthogonal basis.Then its properties and a set of explicit basis expressions with degree k(k=1,2,3) are given.Finally,the new discrete orthogonal base is used to decompose and reconstruct the signal with breakpoints.The experimental results show that this method outperforms the algorithm based on cosine orthogonal basis for expressing the signals with breakpoint.