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| Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials |
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Citation： |
Yingguang SHI.Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials[J].Chinese Annals of Mathematics B,2012,33(5):751~766 |
| Page view： 1828
Net amount： 1153 |
Authors： |
Yingguang SHI; |
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| Abstract： |
In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight $w$ on $\mathbf I=(a,b)$, a function $G\in \mathbf S(w)\!:=\{f\!:\!\int_{\mathbf I}|f(x)|w(x)\rmd x<\infty\}$ satisfying the conditions $G^{(2j)}(x)\ge 0,\ x\in(a,b),\ j=0,1,\cdots,$ and growing as fast as possible as $x\to a+$ and $x\to b-$, plays an important role. But to find such a function $G$ is often difficult and complicated. This implies that to prove convergence of Gaussian quadrature formulas, it is enough to find a function $G\in \mathbf S(w)$ with $G\ge 0$ satisfying $$ \sup_n\sum_{k=1}^n\lambda_{0kn}G(x_{kn})<\infty $$ instead, where the $x_{kn}$'s are the zeros of the $n$th power orthogonal polynomial with respect to the weight $w$ and $\lambda_{0kn}$'s are the corresponding Cotes numbers. Furthermore, some results of the convergence for Gaussian quadrature formulas involving the above condition are given. |
Keywords： |
Convergence, Gaussian quadrature formula, Freud weight |
Classification： |
42C05, 41A55 |
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