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.

Bibliographie

[1] Dzyadyk V. K., “K voprosu o nailuchshem trigonometricheskom priblizhenii kratno monotonnykh funktsii v metrike”, Issledovaniya po sovr. probl. konstr. teorii f, M., 1961, 72–82 | Zbl

[2] Stechkin S. B., “O nailuchshem priblizhenii nekotorykh klassov periodicheskikh funktsii trigometricheskimi polinomami”, Izv. AN SSSR, Ser. matem., 20:5 (1956), 643–648 | MR | Zbl

[3] Telyakovskii S. A., “O normakh trigonometricheskikh polinomov i priblizhenii differentsiruemykh funktsii lineinymi srednimi ikh ryadov Fure”, Tr. MMO, 52 (1961), 61–97

[4] Rusak V. N., “Pryamye metody v ratsionalnoi approksimatsii periodicheskikh funktsii.”, Teoriya funktsii i priblizhenii, ch. 1, Izd-vo Saratovskogo un-ta, 1963, 163–168

[5] Rusak V. N., “Tochnye poryadki nailuchshikh ratsionalnykh priblizhenii na klassakh funktsii, predstavimykh v vide svertki”, DAN SSSR, 279:4 (1984), 810–812 | MR | Zbl

[6] Gonchar A. A., “O skorosti ratsionalnoi approksimatsii nepreryvnykh funktsii s kharakternymi osobennostyami”, Matem. sb., 73(115) (1967), 630–638 | Zbl

[7] Popov V. A., “Ratsionalnaya ravnomernaya approksimatsiya klassa i ee primeneniya”, Dokl. BolgAN, 29:6 (1976), 1–4

[8] Zigmund A., Trigonometricheskie ryady, t. II, Mir, M., 1965 | MR

[9] Rusak V. N., “O priblizhenii ratsionalnymi operatorami tipa Fure periodicheskikh funktsii, predstavimykh v vide svertki”, Izv. AN BSSR. Ser. fiz.-mat., 1984, no. 2, 25–32 | MR | Zbl

[10] Rusak V. N., Ratsionalnye funktsii kak apparat priblizheniya, izd-vo BGU, Minsk, 1979 | MR

[11] Dzhrbashyan M. M., “K teorii ryadov Fure po ratsionalnym funktsiyam”, Izv. AN ArmSSR. Ser. fiz.-mat., 9:7 (1956), 3–28 | MR

[12] Kocharyan T. S., “O priblizhenii ratsionalnymi funktsiyami v kompleksnoi oblasti”, Izv. AN ArmSSR. Ser. fiz.-mat., 11:4 (1958), 11–17

[13] Rusak V. N., “Ob odnom metode priblizheniya ratsionalnymi funktsiyami na veschestvennoi osi”, Matem. zametki, 22:3 (1977), 375–380 | MR | Zbl