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

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 w2 ρ;x of degree n for w2 ρ x . In this paper for an even integer ν ≥ 2 we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejer interpolation polynomials and related approximation process based at the zeros {xk,n,ρ}k 1 of pn w2 ρ;x . Moreover, for an odd integer ν ≥ 1, we give a certain divergence theorem with respect to the higher-order Hermite-Fejer interpolation polynomials based at the zeros {xk,n,ρ}k 1 of pn w2 ρ;x .