Fast algorithms for discrete polynomial transforms on arbitrary grids Original Research Article

Consider the Vandermonde-like matrix P:=(Pk(xM,l))l,k=0M,N, where the polynomials Pk satisfy a three-term recurrence relation and xM,lset membership, variant[−1,1] are arbitrary nodes. If Pk are the Chebyshev polynomials Tk, then P coincides with A:=(Tk(xM,l))l=0,k=0M,N. This paper presents a fast algorithm for the computation of the matrix–vector product Pa in image arithmetical operations. The algorithm divides into a fast transform which replaces Pa with Aã and a fast cosine transform on arbitrary nodes (NDCT). Since the first part of the algorithm was considered in [Math. Comp. 67 (1998) 1577], we focus on approximative algorithms for the NDCT. Our considerations are completed by numerical tests.