Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials

Abstract EN

Let wλ(x):=(1-x2)λ-1/2 and Pλ,n be the ultraspherical polynomials with respect to wλ(x).

Then, we denote the Stieltjes polynomials Eλ,n+1 with respect to wλ(x) satisfying ∫-11wλxPλ,nxEλ,n+1xxmdx =0, 0≤m

In this paper, we consider the higher-order Hermite-Fejér interpolation operator Hn+1,m based on the zeros of Eλ,n+1 and the higher order extended Hermite-Fejér interpolation operator ℋ2n+1,m based on the zeros of Eλ,n+1Pλ,n.

When m is even, we show that Lebesgue constants of these interpolation operators are O(nmax{(1-λ)m-2,0})(0<λ<1) and Onmax1-2λm-2,00<λ<1/2, respectively; that is, ℋ2n+1,m=O(nmax{(1-2λ)m-2,0})(0<λ<1) and Hn+1,m=Onmax1-λm-2,00<λ<1/2.

In the case of the Hermite-Fejér interpolation polynomials ℋ2n+1,m[·] for 1/2≤λ<1, we can prove the weighted uniform convergence.

In addition, when m is odd, we will show that these interpolations diverge for a certain continuous function on [-1,1], proving that Lebesgue constants of these interpolation operators are similar or greater than log n.