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Regular version of the site

Article

A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators

St Petersburg Mathematical Journal. 2016. Vol. 217. No. 1. P. 3-22.
Vinogradov O. L., Gladkaya A.

In what follows, $\sigma>0$, $m,r\in\mathbb N$, $m\geqslant r$, ${\mathbf S}_{\sigma, m}$ is the space of splines of order~$m$ and minimal defect with nodes $\frac{j\pi}{\sigma}$ ($j\in\mathbb Z$), $A_{\sigma,m}(f)_{p}$ is the best approximation of a function~$f$ by the set ${\mathbf S}_{\sigma,m}$ in the space $L_p(\mathbb R)$. It is known that for $p=1,+\infty$

$$

\supl_{f\in W^{(r)}_{p}(\mathbb R)}

\frac{A_{\sigma,m}(f)_{p}}{\|f^{(r)}\|_{p}}=

\frac{{\mathcal K}_r}{\sigma^r}.\eqno(1)

$$

In this paper we construct linear operators ${\mathcal X}_{\sigma,r,m}$ with their values in ${\mathbf S}_{\sigma,m}$, such that for all $p\in[1,+\infty]$ and $f\in W_p^{(r)}(\mathbb R)$

$$

\|f-{\mathcal X}_{\sigma,r,m}(f)\|_{p}\leqslant

\frac{{\mathcal K}_r}{\sigma^r}\|f^{(r)}\|_p.

$$

So we establish the possibility to achieve the upper bounds in~(1) by linear methods of approximation, which was unknown before.