Zhensikbaeb, K. S. Chebyshev monosplines and best quadrature formula. (Russian) Zbl 0703.41017 Vestn. Akad. Nauk Kaz. SSR 1988, No. 4, 73-76 (1988). 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 Keywords:Pólya condition; best renewal; monospline PDFBibTeX XMLCite \textit{K. S. Zhensikbaeb}, Vestn. Akad. Nauk Kaz. SSR 1988, No. 4, 73--76 (1988; Zbl 0703.41017)