| The Convergence of Gr\"{u}nwald Interpolation Operator on the Zeros of Freud Orthogonal Polynomials |
| Received:July 18, 2005 Revised:December 14, 2005 |
| Key Words:
exponential weight orthogonal polynomial interpolation convergence.
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| Fund Project:Open Funds (No.PCN0613) of State Key Laboratory of Oil and Gas Reservoir and Exploitation (Southwest Petroleum University); the Foundation of Education of Zhejiang Province (No.Kyg091206029). |
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| 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 $|f(x) |= O(\exp(\frac{1}{2}|x|^{\beta}))$. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2008.02.013 |
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