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Chebyshev monosplines and best quadrature formula. (Russian) Zbl 0703.41017

Summary: Let L be linear differential operator of order r with constant real coefficients on [a,b] that satisfies the Pólya condition. L defines the class \[ K^ r_ q=\{f\in C^{r-1}[a,b]:\;f^{(r-1)}\quad absolutely\quad continuous\text{ and } \| Lf\|_ q\leq 1\}\quad (1\leq q\leq \infty). \] The problem on best renewal of the integral \(I(f)=\int^{b}_{a}f(x)dx\) within \(K^ r_ q\) (r\(\in N\), \(1\leq q\leq \infty)\) is written in compact form. The renewal uses information \(\lambda =\{\ell_ 1(f),...,\ell_ k(f)\}\) on the values of linear operators \(\ell_ i\) in f. The term \(M^ r_ N\) denotes all L- monosplines with not more than N nodes and \(M_ N^{r0}\) denotes a subset with special integral representation and zero values and derivatives in a and b.The article mainly states that \(M^ r_ N\) contains a unique L-monospline \(m_ p\) with minimal \(L_ p\)-norm that belongs to \(M_ N^{r0}\). The nodes of element \(m_ p\) are pairwise different and the coefficients are positive. A short idea is given on the proof of the statement.

MSC:

41A15 Spline approximation
41A55 Approximate quadratures
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