What distributions of points are possible for convergent sequences of interpolatory integration rules!
Suppose that for \(n\geq 1\), \(I_ n[f]=\sum^ n_{s=1}\omega_{jn}f(x_{jn})\) is an interpolatory quadrature formula so that \(I_ n[P]=\int^ 1_{-1}P(x)dx\) for all \(P\in\pi_{n-1}\). Furthermore, if we suppose that \(\lim_{n\to\infty}In[f]=\int^ 1_{- 1}f(x)dx\) for each \(f\in C[-1,1]\), then in all the classical examples, the nodes \(\{x_{jn}\}\) are found to have arc sine distribution. The authors show that this is not the general case. They show that the set of all possible limit distributions has the form \({1\over 2}[v(x)dx+dv(x)]\) where \(\nu\) is an arbitrary probability measure on \([-1,1]\) and \(v(x)dx={dx\over\pi\sqrt{1-x^ 2}}\). To formulate their main theorem, they denote by \([-1,1]\) the set of all nonnegative Borel measure \(\mu\) on \([-1,1]\) with \(\mu[-1,1]=1\). \(\mu\) is called a probability measure and is also denoted by \(d\mu\). Given a sequence \(\{\nu_ n\}^ \infty_ 1\subset{\mathcal M}[-1,1]\) and \(\nu\in{\mathcal M}[-1,1]\), \(d\nu\) is called a weak limit of a subsequence of \(\{d\nu_ n\}^ \infty_ 1\) iff there exists an infinite subsequence of positive integers such that \(d\nu_ n@>*>>d\nu\) as \(n\to\infty\), \(n\in{\mathcal N}\). In other words \(\forall f\in C[-1,1]\) \[ \lim_{n\to\infty,m\in{\mathcal N}}\int^ 1_{- 1}f(x)d_{\nu_ n}(x)=\int^ 1_{-1}f(x)d\nu(x). \] The main result of the authors is: Theorem 1. (1) Let \(\{I_ n\}^ \infty_ 1\) be a sequence of interpolatory integration rules with \(\lim_{n\to\infty}\| I_ n\|^{1/n}=1\), \(\| I_ n\|=\sum^ n_{j=1}|\omega_{jn}|\), then every weak limit of a subsequence of the corresponding measures \(\{d\mu_ n\}^ \infty_ 1\) has the form \({1\over 2}[d\nu(x)+v(x)dx]\) for some \(\nu\in M[-1,1]\). (2) Conversely, given any \(\nu\in{\mathcal M}[-1,1]\), there exists a sequence \(\{I_ n\}_ 1^ \infty\) of interpolatory quadrature rules with all positive weights, (i.e., \(\omega_{jn}\geq 0\), \(1\leq j\leq n)\) such that the corresponding measures \(\{d\mu_ n\}^ \infty_ 1\) satisfy \[ d\mu_ n@>*>>{1\over 2}(d\nu(x)+\nu(x)dx)\quad\text{as } n\to\infty. \] The authors show that a similar result holds when the quadrature rules have precision \(m(n)\) other than \(n-1\). The paper is well written and makes interesting reading.
scientific article; zbMATH DE number 146901
scientific article