Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

Abstract EN

Let R=(-∞,∞), and let wρ(x)=|x|ρe-Q(x), where ρ>-1/2 and Q∈C1(R):R→R+=[0,∞) is an even function.

Then we can construct the orthonormal polynomials pn(wρ2;x) of degree n for wρ2(x).

In this paper for an even integer ν≥2 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 {xk,n,ρ}k=1n of pn(wρ2;x).

Moreover, for an odd integer ν≥1, we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros {xk,n,ρ}k=1n of pn(wρ2;x).