Orthonormal polynomials for generalized Freud-type weights and higher-order Hermite–Fejér interpolation polynomials

Abstract

Let Q : R-R be even, nonnegative and continuous, Q 0 be continuous, Q 0 40 in ð0; NÞ; and let Q 00 be continuous in ð0; NÞ: Furthermore, Q satisfies further conditions. We consider a certain generalized Freud-type weight W 2 rQ ðxÞ ¼ jxj 2r expðÀ2QðxÞÞ: In previous paper (J. Approx. Theory 121 (2003) 13) we studied the properties of orthonormal polynomials fP n ðW 2 rQ ; xÞg N n¼0 with the generalized Freud-type weight W 2 rQ ðxÞ on R: In this paper we treat three themes. Firstly, we give an estimate of P n ðW 2 rQ ; xÞ in the L p -space, 0oppN: Secondly, we obtain the Markov inequalities, and third we study the higher-order Hermite-Feje´r interpolation polynomials based at the zeros fx kn g n k¼1 of P n ðW 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-Feje´r interpolation polynomials. r 2004 Elsevier Inc. All rights reserved.

Hermite-Fejér and Grünwald interpolation at generalized Laguerre zeros

“…An important property of the weight function u is the following: for every polynomial P m of degree at most m (P m ∈ IP m ) the inequalities [9][10][11][12]:…”

“…and the Bernstein inequality [11], we obtain √ a m m m τ P M ϕu [θa m ,+∞) ≤ C √ a m m m τ e −Am P M ϕu ≤ Cm τ e −Am P M u ≤ Ce −Am P M uand, thenT * m (w, (1 − χ)P M )u ≤ Ce −Am P M u easily follows.Now we can prove Theorem 3.2.Proof. [Proof of Theorem 3.2] Denoting byP N ∈ IP N , N = M log M , M = θm1+θ , the polynomial of best approximation of f ∈ Cū, we can writef − F m (w, f ) = f − P N + H m (w, P N ) − F m (w, f ) = f − P N + F m (w, P N − f ) + T m (w, P N ) + H m (w, (1 − χ)P N ),using Lemma 3.1 and Proposition 4.1, we get( f − F m (w, f ))u ≤ C ( f − P N )ū + √ a N N P N ϕū + e −Am P Nū .Recalling(10) we have( f − P N )ū ≤ Cω ϕ f,Moreover, since (see[15] for a similar argument) √ a N N P N ϕū ≤ Cω ϕ f, fū , the theorem follows.Proof. [Proof of Theorem 3.3] By (3)-(4) and (18) we have f − G m (w, f ) = [ f − F m (w, f )] +F m (w, f ), whereF m (w, f ) = x 1 ≤x k ≤θa m (x)v k (x) f (x k ).…”

A Lagrange-type projector on the real line

“…These polynomials, sometimes called generalized Freud polynomials, have been extensively studied in [5,13,28,6,7].…”

On the Hermite–Fejér Interpolation Based at the Zeros of Generalized Freud Polynomials

“…Szabó in [16]. Recently in [5,6,3,4] the case α ̸ = 0 has been studied assuming that the interpolated functions are uniformly continuous in IR. However this represents a strong limitation.…”