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| ON CONVERGENCE OF PAL-TYPE INTERPOLATION POLYNOMIALS |
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Citation： |
Xie Tingfan.ON CONVERGENCE OF PAL-TYPE INTERPOLATION POLYNOMIALS[J].Chinese Annals of Mathematics B,1988,9(3):315~321 |
| Page view： 744
Net amount： 888 |
Authors： |
Xie Tingfan; |
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| Abstract： |
Let $\[\{ x_k^*\} _{k = 1}^{n - 1}\]$ be the zeros of the (n-1)-th Legendre polynomial $\[{p_{n - 1}}(x)\]$ and $\[\{ {x_k}\} _{k = 1}^n\]$ be the zeros of the polynomial $\[w(x) = (1 - {x^2})p_{n - 1}^1(x)\]$. By the theory of the Pal interpolation，for a function $\[f \in C_{[ - 1,1]}^1\]$, there exists a unique polynomial $\[{Q_n}(f,x)\]$ of degree 2n-l satisfying conditions $\[{Q_n}(f,{x_k}) = f({x_k}),Q_n^'(f,x_k^*) = {f^'}(x_k^*)\]$, where $\[k = 1,2, \cdots ,n\]$ and $\[x_n^* = - 1\]$.The main result of this paper is that if $$\[f \in C_{[ - 1,1]}^r\]$$, then
$$\[f(x) - {Q_n}(f,x) = O(1)W(x)w({f^{(r)}},\frac{1}{n}){n^{\frac{1}{2} - r}}, - 1 \le x \le 1\]$$
Hence, if $\[f \in C_{[ - 1,1]}^1\]$, then $\[{Q_n}(f,x)\]$ converges to the function $\[f(x)\]$ uniformly on the interval $\[[ - 1,1]\]$. |
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