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

Let ℝ = ( − ∞ , ∞ ) , and let 𝑤 𝜌 ( 𝑥 ) = | 𝑥 | 𝜌 𝑒 − 𝑄 ( 𝑥 ) , where 𝜌 > − 1 / 2 and 𝑄 ∈ 𝐶 1 ( ℝ ) ∶ ℝ → ℝ + = [ 0 , ∞ ) is an even function. Then we can construct the orthonormal polynomials 𝑝 𝑛 ( 𝑤 2 𝜌 ; 𝑥 ) of degree 𝑛 for 𝑤 2 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-Fejér interpolation polynomials and related approximation process based at the zeros { 𝑥 𝑘 , 𝑛 , 𝜌 } 𝑛 𝑘 = 1 of 𝑝 𝑛 ( 𝑤 2 𝜌 ; 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 { 𝑥 𝑘 , 𝑛 , 𝜌 } 𝑛 𝑘 = 1 of 𝑝 𝑛 ( 𝑤 2 𝜌 ; 𝑥 ) .