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Convergence and divergence of higher-order Hermite or Hermite-Fejér interpolation polynomials with exponential-type weights. (English) Zbl 1241.41002
Summary: Let =(-,), and let w ρ (x)=|x| ρ e -Q(x) , where ρ>-1/2 and QC 1 (): + =[0,) is an even function. Then we can construct the orthonormal polynomials p n (w ρ 2 ;x) of degree n for w ρ 2 (x). In this paper for an even integer v2 we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and related approximation process based at the zeros {x k,n,ρ } k=1 n of p n (w ρ 2 ;x). Moreover, for an odd integer v1, we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros {x k,n,ρ } k=1 n of p n (w ρ 2 ;x).
MSC:
41A05Interpolation (approximations and expansions)
42C05General theory of orthogonal functions and polynomials