在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性

设$W_{\beta}(x)=\exp(-\frac{1}{2}|x|^{\beta})~(\beta > 7/6)$ 为Freud权, Freud正交多项式定义为满足下式$\int_{- \infty}^{\infty}p_{n}(x)p_{m}(x)W_{\beta}^{2}(x)\rd x=\left \{ \begin{array}{ll} 0 & \hspace{3mm} n \neq m , \\ 1 & \hspace{3mm}n = m \end{array} \right.$的

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

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 $|f(x) |= O(\exp(\frac{1}{2}|x|^{\beta}))$.