赵易,周颂平.在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性[J].数学研究及应用,2008,28(2):340~346 |
在Freud正交多项式零点处的Gr\"{u}nwald插值算子的收敛性 |
The Convergence of Gr\"{u}nwald Interpolation Operator on the Zeros of Freud Orthogonal Polynomials |
投稿时间:2005-07-18 修订日期:2005-12-14 |
DOI:10.3770/j.issn:1000-341X.2008.02.013 |
中文关键词: 指数型权 正交多项式 插值 收敛性. |
英文关键词:exponential weight orthogonal polynomial interpolation convergence. |
基金项目:西南石油大学国家重点实验室开放基金项目(No.PCN0613); 浙江省教育厅项目(No.Kyg091206029). |
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中文摘要: |
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 |
英文摘要: |
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 $|f(x) |= O(\exp(\frac{1}{2}|x|^{\beta}))$. |
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