Geodesic
On Estimates of Uniform Approximations by Rational Fourier--Chebyshev Integral Operators for a Certain Choice of Poles
Matematičeskie zametki, Tome 113 (2023) no. 6, pp. 876-894.

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

The rational Fourier–Chebyshev integral operator with specially chosen poles is considered on the closed interval $[-1,1]$. With the help of the previously obtained upper bound for the uniform approximations of the functions $|x|^s$, $s>0$, on the closed interval $[-1,1]$ by means of the method of rational approximation in use, an asymptotic representation of the corresponding majorant under some conditions on the poles of the approximating function is obtained. To solve this problem, a method has been developed that is based on the classical Laplace method of studying the asymptotic behavior of integrals. The case of modified “Newman parameters” is studied in detail. The values of these parameters are found for which the highest rate of uniform approximations is ensured. In this case, the orders of uniform rational approximations turn out to be higher than those for the corresponding polynomial analogs.
Keywords: rational approximation, integral operators, uniform approximations, asymptotic estimates, Laplace method, Newman parameters, a function with a power-law singularity. 
@article{MZM_2023_113_6_a7,
     author = {P. G. Potseiko and Y. A. Rovba},
     title = {On {Estimates} of {Uniform} {Approximations} by {Rational} {Fourier--Chebyshev} {Integral} {Operators} for a {Certain} {Choice} of {Poles}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {876--894},
     publisher = {mathdoc},
     volume = {113},
     number = {6},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2023_113_6_a7/}
}
TY  - JOUR
AU  - P. G. Potseiko
AU  - Y. A. Rovba
TI  - On Estimates of Uniform Approximations by Rational Fourier--Chebyshev Integral Operators for a Certain Choice of Poles
JO  - Matematičeskie zametki
PY  - 2023
SP  - 876
EP  - 894
VL  - 113
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2023_113_6_a7/
LA  - ru
ID  - MZM_2023_113_6_a7
ER  - 
%0 Journal Article
%A P. G. Potseiko
%A Y. A. Rovba
%T On Estimates of Uniform Approximations by Rational Fourier--Chebyshev Integral Operators for a Certain Choice of Poles
%J Matematičeskie zametki
%D 2023
%P 876-894
%V 113
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2023_113_6_a7/
%G ru
%F MZM_2023_113_6_a7
P. G. Potseiko; Y. A. Rovba. On Estimates of Uniform Approximations by Rational Fourier--Chebyshev Integral Operators for a Certain Choice of Poles. Matematičeskie zametki, Tome 113 (2023) no. 6, pp. 876-894. http://geodesic.mathdoc.fr/item/MZM_2023_113_6_a7/

[1] S. Bernstein, “Sur la meilleure approximation de $|x|$ par des polynomes de degrés donnés”, Acta Math., 37:1 (1914), 1–57 | DOI | MR

[2] R. S. Varga, A. J. Carpenter, “On the Bernstein conjecture in approximation theory”, Constr. Approx., 1:4 (1985), 333–348 | DOI | MR

[3] D. I. Newman, “Rational approximation to $|x|$”, Michigan Math. J., 11 (1964), 11–14 | DOI | MR

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

[5] N. S. Vyacheslavov, “O ravnomernom priblizhenii $|x|$ ratsionalnymi funktsiyami”, Dokl. AN SSSR, 220:3 (1975), 512–515 | MR | Zbl

[6] G. Shtal, “Nailuchshie ravnomernye ratsionalnye approksimatsii $|x|$ na $[-1,1]$”, Matem. sb., 183:8 (1992), 85–118 | MR | Zbl

[7] S. N. Bernshtein, “O nailuchshem priblizhenii $|x|^p$ pri pomoschi mnogochlenov vesma vysokoi stepeni”, Izv. AN SSSR. Ser. matem., 2:2 (1938), 169–190 | Zbl

[8] S. M. Nikolskii, “O nailuchshem priblizhenii mnogochlenami v srednem funktsii $|a-x|^s$”, Izv. AN SSSR. Ser. matem., 11:2 (1947), 139–180 | MR | Zbl

[9] R. A. Raitsin, “Asimptoticheskie svoistva ravnomernykh priblizhenii funktsii s algebraicheskimi osobennostyami chastichnymi summami ryada Fure–Chebysheva”, Izv. vuzov. Matem., 1980, no. 3, 45–49 | MR | Zbl

[10] M. I. Ganzburg, “The Bernstein constant and polynomial interpolation at the Chebyshev nodes”, J. Approx. Theory, 119:2 (2002), 193–213 | DOI | MR

[11] M. Revers, “On the asymptotics of polynomial interpolation to $|x|^\alpha$ at the Chebyshev nodes”, J. Approx. Theory, 165:1 (2013), 70–82 | DOI | MR

[12] A. A. Gonchar, “O skorosti ratsionalnoi approksimatsii nepreryvnykh funktsii s kharakternymi osobennostyami”, Matem. sb., 73 (115):4 (1967), 630–638 | MR | Zbl

[13] G. Freud, J. Szabados, “Rational approximation to $x^\alpha$”, Acta Math. Acad. Sci. Hungar., 18:4 (1967), 393–399 | DOI | MR

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

[15] T. Ganelius, “Rational approximation to $x^\alpha$ on $[0,1]$”, Anal. Math., 5:1 (1979), 19–33 | DOI | MR

[16] J.-E. Andersson, “Rational approximation to functions like $x^\alpha$ in integral norms”, Anal. Math., 14:1 (1988), 11–25 | DOI | MR

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

[18] H. Stahl, “Best uniform rational approximation of $x^\alpha$ on $[0,1]$”, Bull. Amer. Math. Soc. (N.S.), 28:1 (1993), 116–122 | DOI | MR

[19] A. A. Pekarskii, “Approksimatsiya funktsii $z^{\alpha}$ ratsionalnymi drobyami v oblasti s nulevym vneshnim uglom”, Matem. zametki, 91:5 (2012), 761–772 | DOI | MR

[20] S. Takenaka, “On the orthogonal functions and a new formula of interpolations”, Japanese J. of Math., 2 (1925), 129–145 | DOI

[21] F. Malmquist, “Sur la détermination d'une classe de fonctions analytiques par leurs dans un ensemble donné de points”, Compte Rendus Sixiéme Congrés Math. Scand, Kopenhagen, Denmark, 1925, 253–259

[22] M. M. Dzhrbashyan, “K teorii ryadov Fure po ratsionalnym funktsiyam”, Izv. AN Arm. SSR. Ser. fiz.-matem. nauk, 9:7 (1956), 3–28

[23] G. S. Kocharyan, “O priblizhenii ratsionalnymi funktsiyami v kompleksnoi oblasti”, Izv. AN Arm. SSR. Ser. fiz.-matem. nauk, 11:4 (1958), 53–77

[24] A. A. Kitbalyan, “Razlozheniya po obobschennym trigonometricheskim sistemam”, Izv. AN Arm. SSR. Ser. fiz.-matem. nauk, 16:6 (1963), 3–24 | MR

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

[26] P. P. Petrushev, B. A. Popov, Rational Approximation of Real Functions, University Press, Cambridge, 1987 | MR

[27] Y. Rouba, P. Patseika, K. Smatrytski, “On a system of rational Chebyshev–Markov fractions”, Anal. Math., 44:1 (2018), 115–140 | DOI | MR

[28] E. A. Rovba, P. G. Potseiko, “Approksimatsiya funktsii $|x|^s$ na otrezke $[-1,1]$ chastichnymi summami ratsionalnogo ryada Fure–Chebysheva”, Vest. Grodzenskogo un-ta, 9:3 (2019), 16–28

[29] E. A. Rovba, P. G. Potseiko, “O ratsionalnoi approksimatsii funktsii markovskogo vida chastichnymi summami ryadov Fure po odnoi sisteme Chebysheva–Markova”, Matem. zametki, 108:4 (2020), 572–587 | DOI | MR

[30] M. A. Evgrafov, Asimptoticheskie otsenki i tselye funktsii, Nauka, M., 1979 | MR

[31] M. V. Fedoryuk, Asimptotika: integraly i ryady, Nauka, M., 1987 | MR

[32] E. A. Rovba, “Ob odnom pryamom metode v ratsionalnoi approksimatsii”, Dokl. AN BSSR, 23:11 (1979), 968–971 | MR

[33] P. G. Potseiko, E. A. Rovba, “Priblizheniya na klassakh integralov Puassona ratsionalnymi integralnymi operatorami Fure–Chebysheva”, Sib. matem. zhurn., 62:2 (2021), 362–386 | DOI | MR

[34] Yu. V. Sidorov, M. V. Fedoryuk, M. I. Shabunin, Lektsii po teorii funktsii kompleksnogo peremennogo, Nauka, M., 1989 | MR

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