Rational approximations of absolutely continuous functions with derivative in an Orlicz space

A. A. Pekarskii

Sbornik. Mathematics, Tome 45 (1983) no. 1, pp. 121-137

Résumé

Let $R_n(f)$ be the best uniform approximation of $f \in C[0,1]$ by rational fractions of degree at most $n$, and let $ W[0,1]$ be the set of monotone convex functions $w\in C[0,1]$ such that $w(0)=0$ and $w(1)=1$. Theorem 1. Suppose the function $f$ is absolutely continuous on the interval $[0,1],$ and let $w\in W[0,1]$ and $\widehat f= f(w(x))$. If $|\widehat f'|\ln^+|\widehat f'|$ is summable on $[0,1],$ then $R_n(f)=o(1/n)$. Various applications and generalizations of this result are given, and the periodic case is also considered. Bibliography: 23 titles.

Bibliographie

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

[2] Gonchar A. A., Skorost priblizheniya ratsionalnymi drobyami i svoistva funktsii (Trudy Mezhdunarodnogo kongressa matematikov, 1966 g.), Mir, M., 1968, s. 329–356

[3] Popov V. A., Szabados J., “A remark on the rational approximation of functions”, C. r. Acad. Bulgare Sci., 28:5 (1975), 1303–1306 | MR

[4] Pekarskii A. A., Ratsionalnaya approksimatsiya i prostranstva Orlicha, Dep. v VINITI per. No 314-78 (RZhMat., 1978, 7B730), 28 s

[5] Dolzhenko E. P., “Ravnomernye approksimatsii ratsionalnymi funktsiyami (algebraicheskimi i trigonometricheskimi) i globalnye funktsionalnye svoistva”, DAN SSSR, 166:3 (1966), 526–529 | Zbl

[6] Gonchar A. A., “O nailuchshikh priblizheniyakh ratsionalnymi funktsiyami”, DAN SSSR, 100:2 (1955), 205–208 | Zbl

[7] Pekarskii A. A., “Ratsionalnye priblizheniya absolyutno nepreryvnykh funktsii s proizvodnoi iz prostranstva Orlicha”, DAN BSSR, 24:4 (1980), 301–304 | MR | Zbl

[8] Pekarskii A. A., “Ratsionalnaya approksimatsiya nepreryvnykh funktsii s zadannymi modulem nepreryvnosti i modulem izmeneniya”, Izv. AN BSSR, ser. fiz.-matem. nauk, 1978, no. 5, 34–39 | MR | Zbl

[9] Markushevich A. I., Teoriya analiticheskikh funktsii, T. 1, Nauka, M., 1967

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

[11] Bekkenbakh E., Bellman R., Neravenstva, Mir, M., 1965 | MR

[12] Gonchar A. A., “O zadachakh E. I. Zolotareva, svyazannykh s ratsionalnymi funktsiyami”, Matem. sb., 78 (120) (1969), 640–654 | Zbl

[13] Zigmund A., Trigonometricheskie ryady, T. 1, Mir, M., 1965 | MR

[14] Pekarskii A. A., “Ratsionalnaya approksimatsiya absolyutno nepreryvnykh funktsii”, Izv. AN BSSR, ser. fiz.-matem. nauk, 1978, no. 6, 22–26 | MR | Zbl

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

[16] Chanturiya Z. A., “Modul izmeneniya funktsii i ego primenenie v teorii ryadov Fure”, DAN SSSR, 214:1 (1974), 63–66 | Zbl

[17] Bulanov A. P., “Ratsionalnye priblizheniya nepreryvnykh funktsii s konechnym izmeneniem”, Izv. AN SSSR, ser. matem., 39 (1975), 1142–1181 | MR | Zbl

[18] Dzyadyk V. K., Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977 | MR | Zbl

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

[20] Popov V. A., “Uniform rational approximation of the classe $V_r$ and its applications”, Acta Math., 29:1–2 (1977), 119–129 | MR | Zbl

[21] Pekarskii A. A., “Metod posledovatelnykh usrednenii v teorii ratsionalnoi approksimatsii”, DAN BSSR, 21:10 (1977), 876–878 | MR | Zbl

[22] Pekarskii A. A., “Ratsionalnaya approksimatsiya singulyarnykh funktsii”, Izv. AN BSSR, ser. fiz.-matem. nauk, 1980, no. 3, 32–40 | MR | Zbl

[23] Petrushev P. P., “Ravnomernye ratsionalnye approksimatsii funktsii s konechnym izmeneniem”, PLISKA B'lgarski matem. studii, 1 (1977), 145–155 | Zbl