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Sharp order estimates for best rational approximations in classes of functions representable as convolutions
V. N. Rusak
Abstract:
Let
$h(t)$ be a function of bounded variation,
$[\operatorname{Var}h(t)]_0^{2\pi}\leqslant1$,
and
$D_r(t)$ the Weyl kernel of order
$r$, i.e.
$D_r(t)=\sum_{k=1}^\infty k^{-r}\cos\bigl(kt-\frac {r\pi}{2}\bigr)$,
$r>0$.
Denote by
$W_{2\pi}^r V$ and
$W_{2\pi}^r V_0$ the classes of functions represented by the corresponding formulas
$$
f(k)=\frac{a_0}2+\frac1\pi\int_0^{2\pi}D_r(x-t)h(t)\,dt, \qquad f(x)=\frac1\pi\int_0^{2\pi}D_{r+1}(x-t)\,dh(t).
$$
The conjugate classes of functions
$\widetilde{W_{2\pi}^r V}$ and
$\widetilde{W_{2\pi}^r V_0}$ are also considered; they are convolutions of conjugate Weyl kernels with functions of bounded variation.
The following main result is proved:
$$
\sup_{f\in K^r}\mathbf R_n^T(f)\asymp\frac1{n^{r+1}},
$$
where
$\mathbf R_n^T(f)$ is the best uniform approximation by trigonometric rational functions of order at most
$n$, and
$K^r$ is one of the classes
$$
W_{2\pi}^r V,\qquad W_{2\pi}^r V_0,\qquad\widetilde{W_{2\pi}^r V},\qquad\widetilde{W_{2\pi}^r V_0}.
$$
Bibliography: 13 titles.
UDC:
517.51+
517.53
MSC: 41A20,
42A10,
41A25 Received: 21.09.1984
Fulltext:
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