Geodesic
Uniform Rational Approximation of Even and Odd Continuations of Functions
Matematičeskie zametki, Tome 115 (2024) no. 2, pp. 257-265.

Voir la notice de l'article provenant de la source Math-Net.Ru

The behavior of the best rational approximations of an odd continuation of a function is studied. It is shown that without additional conditions on the smoothness of the function, it is impossible to estimate the best rational approximation of the odd continuation of the function on $[-1,1]$ in terms of the best rational approximation of the original function on $[0,1]$. A sharp upper bound is found for the best rational approximations of an even (odd) continuation of a function in terms of an odd (even) continuation and an extremal Blaschke product.
Keywords: rational approximation, best uniform approximation, kink function, odd continuation, even continuation, Blaschke product, power function, function with logarithmic singularity. 
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T. S. Mardvilko. Uniform Rational Approximation of Even and Odd Continuations of Functions. Matematičeskie zametki, Tome 115 (2024) no. 2, pp. 257-265. http://geodesic.mathdoc.fr/item/MZM_2024_115_2_a8/

[1] A. A. Gonchar, “Otsenki rosta ratsionalnykh funktsii i nekotorye ikh prilozheniya”, Matem. sb., 72(114):3 (1967), 489–503 | MR | Zbl

[2] T. S. Mardvilko, A. A. Pekarskii, “Primenenie deistvitelnogo prostranstva Khardi–Soboleva na pryamoi dlya issledovaniya skorosti ravnomernykh ratsionalnykh priblizhenii funktsii”, Zh. Beloruss. gos. un-ta. Matem. Inform., 3 (2022), 16–36 | MR

[3] G. G. Lorenz, M. v. Golitschek, Y. Makovoz, Constructive Approximation. Advanced Problems, Grundlehren Math. Wiss., 304, Springer-Verlag, Berlin, 1996 | MR

[4] H. Stahl, “Best uniform rational approximation of $x^{\alpha}$ on $[0,1]$”, Acta Math., 190:2 (2003), 241–306 | DOI | MR

[5] D. J. Newman, “Rational approximation to $|x|$”, Mich. Math. J., 11:1 (1964), 11–14 | DOI | MR

[6] A. P. Bulanov, “Asimptotika dlya naimenshikh uklonenii $|x|$ ot ratsionalnykh funktsii”, Matem. sb., 76(118):2 (1968), 288–303 | MR | Zbl

[7] J. Tzimbalario, “Rational approximation to $x^{\alpha}$”, J. Approximation Theory, 16:2 (1976), 187–193 | DOI | MR

[8] N. S. Vyacheslavov, “O naimenshikh ukloneniyakh funktsii $\operatorname{sign}x$ i ee pervoobraznykh ot ratsionalnykh funktsii v metrikakh $L_p$, $0

\leqslant\infty$”, Matem. sb., 103(145):1(5) (1977), 24–36 | MR | Zbl

[9] N. S. Vyacheslavov, “Ob approksimatsii $x^\alpha$ ratsionalnymi funktsiyami”, Izv. AN SSSR. Ser. matem., 44:1 (1980), 92–109 | MR | Zbl

[10] A. A. Pekarskii, “Nailuchshie ravnomernye ratsionalnye priblizheniya funktsii Markova”, Algebra i analiz, 7:2 (1995), 121–132 | MR | Zbl

[11] E. V. Kovalevskaya, A. A. Pekarskii, “Postroenie ekstremalnykh proizvedenii Blyashke”, Vesn. GrDU im. Ya. Kupaly. Ser. 2, 7:1 (2017), 6–13 | MR

[12] T. S. Mardvilko, “Approksimatsiya chetnogo i nechetnogo prodolzheniya funktsii”, Sovremennye metody teorii funktsii i smezhnye problemy. Materialy mezhdunarodnoi konferentsii Voronezhskaya zimnyaya matematicheskaya shkola, 2023, 245–247, Voronezh