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Analog of the Akhiezer-Krein-Favard sums for periodic splines of minimal defect. (English. Russian original) Zbl 1050.41016
Let $n,r,m\in\Bbb N$, $m\geq r$, $W^{(r)}_{p}$ be the space of $2\pi$-periodic functions whose $(r - 1)$th-order derivative is absolutely continuous on any segment and $r$th-order derivative belongs to $L_p$, and $S_{2n,m}$ be the space of $2\pi$-periodic splines of order $m$ of minimal defect over the uniform partition $k\pi/n$ ($k\in\Bbb Z$). The author constructs linear operators $X_{n,r,m}:L_1\to S_{2n,m}$ such that $$\sup_{f\in W^{(r)}_{\infty}} \frac{\Vert f-X_{n,r,m}(f)\Vert _\infty}{\Vert f^{(r)}\Vert _\infty}= \sup_{f\in W^{(r)}_1} \frac{\Vert f-X_{n,r,m}(f)\Vert _1}{\Vert f^{(r)}\Vert _1}= \frac{K_r}{n^r},$$ where $$K_r= \dfrac4\pi\sum\limits_{l=0}^{\infty} \dfrac{(-1)^{l(r+1)}}{(2l+1)^{r+1}}.$$ The operators $X_{n,r,m}$ are constructed using the interpolation of Bernoulli kernels. As is proved, the operators $X_{n,r,m}$ converge to polynomial Akhiezer-Krein-Favard operators as $m\to\infty$.
Reviewer: Yu. S. Volkov (Novosibirsk)

41A35Approximation by operators (in particular, by integral operators)
41A15Spline approximation