Convergence and divergence of higher-order Hermite or Hermite-Fejér interpolation polynomials with exponential-type weights
Summary: Let \(\mathbb R = (-\infty, \infty)\), and let \(w_\rho(x) = |x|^\rho e^{-Q(x)}\), where \(\rho > -1/2\) and \(Q \in C^1(\mathbb R) : \mathbb R \rightarrow \mathbb R^+ = [0, \infty)\) is an even function. Then we can construct the orthonormal polynomials \(p_n(w^2_\rho; x)\) of degree \(n\) for \(w^2_\rho(x)\). In this paper for an even integer \(v \geq 2\) 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 \(\{x_{k,n,\rho}\}^n_{k=1}\) of \(p_n(w^2_{\rho}; x)\). Moreover, for an odd integer \(v \geq 1\), we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros \(\{x_{k,n,\rho}\}^n_{k=1}\) of \(p_n(w^2_\rho; x)\).
scientific article; zbMATH DE number 6049793
scientific article