Error estimates of Lagrange interpolation and orthonormal expansions for Freud weights

Abstract

Let S-n[f] be the nth partial sum of the orthonormal polynomials expansion with respect to a Freud weight. Then we obtain sufficient conditions for the boundedness of S-n[f] and discuss the speed of the convergence of S-n[f] in weighted L-p space. We also find sufficient conditions for the boundedness of the Lagrange interpolation polynomial L-n[f], whose nodal points are the zeros of orthonormal polynomials with respect to a Freud weight. In particular, if W(x) = e(-(1/2)alpha2) is the Hermite weight function, then we obtain sufficient conditions for the inequalities to hold: parallel to (S-n[f] - f)((k))Wu(b)parallel to (LP(R)) less than or equal to C (1/rootn)(r-k) parallel tof((r))Wu(B)parallel to (Lp(R)) and parallel to (L-n[f] - f)((k))Wu(b)parallel to (Lp(R)) less than or equal to C (1/rootn)(r-k) parallel tof((r))W(1+x(2))(r/3)u(B)parallel to (LP(R)), where u(gamma)(x) = (1 + \x \)(gamma), gamma epsilon R and k = 0,1,2...,r. (C) 2001 Elsevier Science B.V. All rights reserved.