The Convergence of Gr\"{u}nwald Interpolation Operator on the Zeros of Freud Orthogonal Polynomials

Abstract:

      Let $W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})$ be the Freud weight and $p_{n}(x) \in \Pi_{n}$ be the sequence of orthogonal polynomials with respect to $W_{\beta}^{2}(x)$, that is, $$ \int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0, & \hspace{3mm} n \neq m , \\ 1, & \hspace{3mm}n = m. \end{array} \right.$$ It is known that all the zeros of $p_{n}(x)$ are distributed on the whole real line. The present paper investigates the convergence of Gr\"{u}nwald interpolatory operators based on the zeros of orthogonal polynomials for the Freud weights. We prove that, if we take the zeros of Freud polynomials as the interpolation nodes, then $$G_{n}(f,x) \rightarrow f(x), ~~n \rightarrow \infty$$ holds for every $x \in (-\infty,\infty)$, where $f(x)$ is any continous function on the real line satisfying $|f(x) |= O(\exp(\frac{1}{2}|x|^{\beta}))$.