Sharp order estimates for best rational approximations in classes of functions representable as convolutions

V. N. Rusak

Sbornik. Mathematics, Tome 56 (1987) no. 2, pp. 491-513

Résumé

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.