Orthonormal polynomials for generalized Freud-type weights and higher-order Hermite-Feje'r interpolation polynomials

Let Q : R-R be even, nonnegative and continuous, Q0 be continuous, Q040 in ð0;NÞ; and let Q00 be continuous in ð0;NÞ: Furthermore, Q satisfies further conditions. We consider a certain generalized Freud-type weight W 2 rQðxÞ 1⁄4 jxj 2r expð 2QðxÞÞ: In previous paper (J. Approx. Theory 121 (2003) 13) we studied the properties of orthonormal polynomials fPnðW 2 rQ; xÞg N n1⁄40 with the generalized Freud-type weight W 2 rQðxÞ on R: In this paper we treat three themes. Firstly, we give an estimate of PnðW 2 rQ; xÞ in the Lp-space, 0oppN: Secondly, we obtain the Markov inequalities, and third we study the higher-order Hermite–Fejér interpolation polynomials based at the zeros fxkngnk1⁄41 of PnðW 2 rQ; xÞ: In Section 5 we show that our results are applicable to the study of approximation for continuous functions by the higher-order Hermite-Fejér interpolation polynomials. r 2004 Elsevier Inc. All rights reserved.