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Interpolatory product integration for Riemann-integrable functions. (English) Zbl 0636.41021

Let R be the class of Riemann-integrable functions on [-1,1] and A the set of admissible weight functions w for which \(w\in L_ 1\) and \(w>0\) almost everywhere. The paper is concerned with the approximation of the integral \(I(kf)=\int^{1}_{-1}k(x)f(x)dx,\) \(k\in L_ 1\), \(f\in R\), by integrations rules of the form \(I_ n(f;k)=\sum^{n}_{i=1}w_{in}(k)\) \(f(x_{in})\), \(n=1,2,...\), for which \(I_ n(f;k)=I(kf)\) whenever f is a polynomial of degree \(\leq n-1.\)
One of the results is the following theorem: Let \(\{X_ n\}\), \(X_ n=\{x_{in}:\) \(i=1,...,n\}\), be a sequence of point sets such that, for some \(v\in A\) and every n, the rule \(I_ n(f;v)\) based on \(X_ n\) is exact for all polynomials of degree \(\leq cn\) for some \(c>1\) and all sufficiently large n, and such that \(I_ n(f;v)\to I(vf)\) for all \(f\in R\). Assume that for some \(p,1<p<\infty\), there exists a function \(\rho \in L_ q,\quad p^{-1}+q^{-1}=1,\) such that \(I_ n(f;k)\to I(kf)\) for all \(f\in C\) and all \(k\in L_ p^{(1/\rho)}\). Then \(I_ n(f;k)\to I(kf)\) for all \(f\in R\) and the same k. If in addition \(\sum^{n}_{i=1}| w_{in}(v)| \to I(v),\) in particular, if \(w_{in}(v)>0\), \(i=1,...,n\) for n sufficiently large, then the companion rule \(\to I(| k| f)\) for all \(f\in R\) and the same k. Several applications of the general results are given.
Reviewer: R.Precup

MSC:

41A55 Approximate quadratures
65D32 Numerical quadrature and cubature formulas
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