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\title{Optimal recovery in Hardy--Sobolev spaces
and an analogue of splines for analytic functions}
\author{K.~Yu.~Osipenko, ``MATI" --- Russian State Technology University,
Moscow}
\maketitle

Denote by $H_\infty^r$ the class of analytic in the unit disk $D$ functions
$f$ for which $|f^{(r)}(z)|\le1$, $z\in D$. We consider the problem of
optimal recovery of $f(\tau)$, $\tau\in(-1,1)$, using the information $If=(
f(t_1),\ldots,f(t_{n+r}))$, $t_j\in(-1,1)$. We calculate the value
$$e(\tau,H_\infty^r,I)=\infp_{S\colon\mathbb C^{n+r}\to\mathbb C}\sup_{f\in
H_\infty^r}|f(\tau)-S(If)|$$
and find an optimal algorithm $S_0$ for which the infimum is attained.

For every system of points $-1<t_1<\ldots<t_{n+r}<1$ there exist such $-1<x
_1<\ldots<x_n<1$ that the function
$$g\in X_{n+r}=\spa\{1,z,\ldots,z^{r-1},g_1(z),\ldots,g_n(z)\},$$
where
$$g_j(z)=\int_0^z\frac{(z-t)^{r-1}(1-t^2)\omega_j(t)}{(r-1)!(1-x_jt)^2}\,dt
,\quad\omega_j(t)=\prod_{\substack{k=1\\k\ne j}}^n\frac{t-x_k}{1-x_kt},$$
which interpolates $f$ at the points $t_1,\ldots,t_{n+r}$ gives an optimal
method of recovery $f(\tau)\approx g(\tau)$ for all $\tau\in(-1,1)$. Thus
the space $X_{n+r}$ is an analogue of polynomial splines which appear in
the similar problem for the Sobolev classes.

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