Tchebycheff rational approximation in the disk, on the circle, and on a closed interval

A. A. Pekarskii

Sbornik. Mathematics, Tome 61 (1988) no. 1, pp. 87-102

Résumé

Suppose that the function $f$ is analytic in the disk $\{z:|z|<1\}$ and continuous in its closure. Let $R_n(f)$ denote the best uniform approximation of $f$ by rational functions of degree at most $n$. In 1965 Dolzhenko established that if $\sum R_n(f)<\infty$ then $f'$ belongs to the Hardy space $H_1$. The following converse of this result is obtained here: if $f'\in H_1$, then $R_n(f)=O(1/n)$. In combination with results of Peller, Semmes, and the author, this estimate yields, in particular, a description of the set of functions $f$ with $\bigl[\sum(2^{k\alpha }R_{2^k}(f))^q\bigr]^{1/q}<\infty$, where $\alpha>1$ and $0<q\le\infty$. Bibliography: 38 titles.

Bibliographie

[1] Dolzhenko E. P., “Skorost priblizheniya ratsionalnymi drobyami i svoistvami funktsii”, Matem. sb., 56(98) (1962), 403–433

[2] Gonchar A. A., “Skorost priblizheniya ratsionalnymi drobyami i svoistva funktsii”, Tr. mezhdunarodnogo kongressa matematikov (M., 1966), Mir, M., 1968, 329–346

[3] Dolzhenko E. P., “Ratsionalnye approksimatsii i granichnye svoistva analiticheskikh funktsii”, Matem. sb., 69(111) (1966), 497–524 | Zbl

[4] Pekarskii A. A., O skorosti nailuchshikh ratsionalnykh priblizhenii v prostranstvakh $C$, $BMO$ i $L_p$, Dep. v VINITI, No 6691–84

[5] Pekarskii A. A., Ratsionalnye priblizheniya analiticheskikh v kruge funktsii s proizvodnoi iz prostranstva Khardi $H_1$, Dep. v VINITI, No 4055–85

[6] Coifman R. R., “A real variable characterization of $H^p$”, Stud. Math., 51:3 (1974), 269–274 | MR | Zbl

[7] Kashin B. S, Saakyan A. A., Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[8] Rusak V. N., “O poryadke priblizheniya polozhitelnymi ratsionalnymi operatorami”, Izv. AN BSSR. Ser. fiz.-matem., 1975, 39–46 | MR | Zbl

[9] Rusak V. N., Ratsionalnye funktsii kak apparat priblizheniya, Izd-vo BGU im. V. I. Lenina, Minsk, 1979 | MR

[10] Pekarskii A. A., “Otsenki proizvodnoi integrala tipa Koshi s meromorfnoi plotnostyu i ikh prilozheniya”, Matem. zametki, 31:3 (1982), 389–402 | MR | Zbl

[11] Pekarskii A. A., “Klassy analiticheskikh funktsii, opredelyaemye nailuchshimi ratsionalnymi priblizheniyami v $H_p$”, Matem. sb., 127(169) (1985), 3–20 | MR | Zbl

[12] Brudnyi Yu. A., “Ratsionalnaya approksimatsiya i teoremy vlozheniya”, DAN SSSR, 247:2 (1979), 269–272 | MR | Zbl

[13] Peller V. V., “Opisanie operatorov Gankelya klassa $\sigma_p$ pri $p>0$, issledovanie skorosti ratsionalnoi approksimatsii i drugie prilozheniya”, Matem. sb., 122(164) (1983), 481–510 | MR

[14] Semmes S., “Trace ideal criteria for Hankel operators and applications to Besov spaces”, Integral Equat. and Oper. Theory, 7:2 (1984), 241–281 | DOI | MR | Zbl

[15] Pekarskii A. A., “Neravenstva tipa Bernshteina dlya proizvodnykh ratsionalnykh funktsii i obratnye teoremy ratsionalnoi approksimatsii”, Matem. sb., 124(166) (1984), 571–588 | MR | Zbl

[16] Pekarskii A. A., “O skorosti nailuchshikh ratsionalnykh priblizhenii v prostranstvakh $C$, $BMO$ i $L_p$”, Dokl. rasshir. zasedanii semin. In-ta prikl. matem. im. I. N. Vekua, 1:2 (1985), 106–109 | MR

[17] Danilyuk I. I., Neregulyarnye granichnye zadachi na ploskosti, Nauka, M., 1978

[18] Garnet Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984 | MR | Zbl

[19] Flett T. M., “Lipschitz spaces of functions on the cirkle and the disc”, J. Math. Anal. and Appl., 39:1 (1972), 121–158 | DOI | MR

[20] Peller V. V., “Operatory Gankelya klassa $\sigma_p$ i ikh prilozheniya (ratsionalnaya approksimatsiya, gaussovskie protsessy, problema mazhoratsii operatorov)”, Matem. sb., 113(155) (1980), 538–581 | MR | Zbl

[21] Dynkin E. M., “O klassakh $B_p{}^s$ pri $0

1$”, DAN SSSR, 275:5 (1984), 9–12 | MR

[22] Oswald P., “On Besov–Hardy–Sobolev spaces of analytic function in the unit disk”, Czechosl. Math. J., 33:3 (1983), 408–426 | MR | Zbl

[23] Pekarskii A. A., “Ratsionalnye priblizheniya vypuklykh funktsii”, Matem. zametki., 38:5 (1985), 679–690 | MR | Zbl

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

[25] Gonchar A. A., “Skorost ratsionalnoi approksimatsii i svoistvo odnoznachnosti analiticheskoi funktsii v okrestnosti izolirovannoi osoboi tochki”, Matem. sb., 94(136) (1974), 265–280

[26] Bulanov A. P., “Ratsionalnye priblizheniya kusochno-vypuklykh funktsii”, Konstruktivnaya teoriya funktsii. Tr. mezhdunar. konf. (Varna, 1981), Sofiya, 1983, 150–156

[27] Berg I., Lefstrem I., Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[28] Semmes E., “Another characterization of $H^p$, $0

\infty$, with application to interpolation”, Lect. Notes. Math., 992, 1983, 212–226 | MR

[29] Peller V. V., “Ratsionalnaya approksimatsiya i gladkost funktsii”, 3ap. nauchn. seminarov LOMI, 107 (1982), 150–159 | MR | Zbl

[30] Kakhan Zh.-P., Absolyutno skhodyaschiesya ryady Fure, Mir, M., 1976

[31] Pekarskii A. A., “Ratsionalnye priblizheniya absolyutno nepreryvnykh funktsii s proizvodnoi iz prostranstva Orlicha”, Matem. sb., 117(159) (1982), 114–130 | MR | Zbl

[32] Popov V. A., Petrushev P. P., “Tochnyi poryadok nailuchshego ravnomernogo priblizheniya vypuklykh funktsii ratsionalnymi funktsiyami”, Matem. sb., 103(145) (1977), 285–292 | MR | Zbl

[33] Krasnoselskii M. A., Rutitskii Ya. B., Vypuklye funktsii i prostranstva Orlicha, Fizmatgiz, M., 1958 | MR

[34] Grigoryan L. D., “Otsenki normy golomorfnykh, sostavlyayuschikh meromorfnykh funktsii v oblastyakh s gladkoi granitsei”, Matem. sb., 100(142) (1976), 156–164 | Zbl

[35] Sevastyanov E. A., “Ratsionalnaya approksimatsiya i absolyutnaya skhodimost ryadov Fure”, Matem. sb., 107(149) (1978), 227–244 | MR

[36] Stechkin S. B., “O nailuchshem priblizhenii sopryazhennykh funktsii trigonometricheskimi polinomami”, Izv. AN SSSR. Ser. matem., 20:2 (1956), 197–206 | Zbl

[37] Peller V. V., Khruschev S. V., “Operatory Gankelya, nailuchshie priblizheniya i statsionarnye gaussovskie protsessy”, UMN, 37:1 (1982), 53–124 | MR | Zbl

[38] Dynkin E. M., “Otsenki analiticheskikh funktsii v zhordanovykh oblastyakh”, Zap. nauchn. seminarov LOMI, 73 (1977), 70–90 | MR