On the problem of the description of sequences of best rational trigonometric approximations

A. P. Starovoitov

Sbornik. Mathematics, Tome 191 (2000) no. 6, pp. 927-936

Résumé

For a fixed sequence $\{a_n\}^\infty_{n=0}$ of non-negative real numbers strictly decreasing to zero a continuous $2\pi$-periodic function $f$ is constructed such that $R^T_n(f)=a_n$, $n=0,1,2,\dots$, where the $R^T_n(f)$ are the best approximations of $f$ in the uniform norm by rational trigonometric functions of degree at most $n$.

Bibliographie

[1] Bernstein S., “Sur le probléme inverse de la théorie de la meilleure approximation des fonctions continues”, C. R. Acad. Sci. Paris, 206 (1938), 1520–1523 | Zbl

[2] Natanson I. P., Konstruktivnaya teoriya funktsii, GITTL, M.–L., 1949

[3] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960

[4] Dolzhenko E. P., “Sravnenie skorostei ratsionalnoi i polinomialnoi approksimatsii”, Matem. zametki, 1:3 (1967), 313–320 | MR | Zbl

[5] Lorentz G., v. Golitschek M., Makavoz Y., Consturctive approximation: advanced problems, Springer-Verlag, Berlin, 1996 | MR

[6] Hayashi E., Trefethen L., Gutknecht M., “The CF table”, Constr. Approx., 6:2 (1990), 195–223 | DOI | MR | Zbl

[7] Trefethen L., “Square blocks and equioscillation in the Pade, Walsh and CF tables”, Rational approximation and interpolation, Proc. Conf. (Tampa/Fla, 1983), Lect. Notes in Math., 1105, 1984, 170–181 | MR | Zbl

[8] Levin A. L., Tikhomirov V. M., “O priblizhenii analiticheskikh funktsii ratsionalnymi”, Dokl. AN SSSR, 174:2 (1967), 279–282 | MR | Zbl

[9] Levin A. L., “Priblizhenie ratsionalnymi funktsiyami v kompleksnoi oblasti”, Matem. zametki, 9:2 (1971), 121–130 | MR | Zbl

[10] Boehm B., “Functions, whose best rational Chebyshew approximations are polinomials”, Numer. Math., 6:3 (1964), 235–242 | DOI | MR | Zbl

[11] Pekarskii A. A., “O sluchayakh sovpadeniya nailuchshikh ravnomernykh polinomialnykh i ratsionalnykh priblizhenii”, J. Tech. Univ. at Plovdiv. Fund. Sci. and Appl., 2 (1996), 41–44 | MR

[12] Pekarskii A. A., “Suschestvovanie funktsii s zadannymi nailuchshimi ravnomernymi ratsionalnymi priblizheniyami”, Izv. AN Belarusi. Ser. fiz.-matem. nauk, 1994, no. 1, 23–26 | MR | Zbl

[13] Nazarenko M. A., “Suschestvovanie funktsii s zadannymi ratsionalnymi priblizheniyami v prostranstve $C(A)$”, Vestn. MGU. Ser. 1. Matem., mekh., 1997, no. 4, 20–22 | MR

[14] Bernshtein S. N., Ekstremalnye svoistva polinomov i nailuchshee priblizhenie nepreryvnykh funktsii odnoi veschestvennoi peremennoi, ONTI, M.–L., 1937

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

[16] Chebyshev P. L., Polnoe sobranie sochinenii, T. 2, Izd-vo AN SSSR, M.–L., 1947 | MR

[17] Markov A. A., Izbrannye trudy, GITTL, M.–L., 1948

[18] Bernshtein S. N., Sobranie sochinenii v 4-kh tomakh, T. 1, Izd-vo AN SSSR, M., 1952

[19] Bernshtein S. N., Sobranie sochinenii v 4-kh tomakh, T. 2, Izd-vo AN SSSR, M., 1954

[20] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR

[21] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1977 | MR | Zbl

[22] Starovoitov A. P., “Ob odnoi probleme Bernshteina–Dolzhenko”, International conference on approximation theory and its applications dedicated to the memory of V. K. Dzjadyk, Abstracts, Institute of Math. Nat. Acad. Sci. Ukraine, Kyiv, 1999, 78

[23] Starovoitov A. P., “Opisanie posledovatelnostei nailuchshikh trigonometricheskikh ratsionalnykh priblizhenii”, Analiticheskie metody analiza i differentsialnykh uravnenii, Tez. dokl. mezhdunar. konf. (Belarus, Minsk, 14–18 sentyabrya 1999 g.), BGU, Minsk, 1999, 213—214

[24] Starovoitov A. P., “K probleme opisaniya posledovatelnostei nailuchshikh trigonometricheskikh ratsionalnykh priblizhenii”, Dokl. NAN Belarusi, 44:2 (2000), 23–25 | MR