On a filter for exponentially localized kernels based on Jacobi polynomials

Let alpha, beta >= - 1/2 and for k = 0, 1, ..., p(k)((alpha, beta)) denote the orthonormalized Jacobi polynomial of degree k. k. We discuss the construction of a matrix H so that there exist positive constants c, cl, depending only on H, alpha, and beta such that vertical bar Sigma H-infinity(k=0)k, (n)p(k)((alpha, beta))(cos theta)p(k)((alpha, beta))(cos phi)vertical bar <= c(1)n(2max(alpha, beta)+2) exp(-cn(theta - phi)(2)), theta, phi is an element of [0, pi], n = 1, 2, .... Specializing to the case of Chebyshev polynomials, alpha=beta=-1/2, we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L-2 space. (C) 2009 Elsevier Inc. All rights reserved.