Approximation by Fourier sums in classes of Weyl – Nagy differentiable functions with high exponent of smoothness

  • A. S. Serdyuk Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev
  • I. V. Sokolenko Institute of Mathematics, National Academy of Sciences of Ukraine
Keywords: Fourier sum, Weyl-Nagy class, asymptotic equality

Abstract

UDC 517.5

We establish asymptotic estimates for the least upper bound of approximations in the uniform metric by Fourier sums of order $n-1$ in classes of $2\pi$-periodic Weyl-Nagy differentiable functions $W^r_{\beta,p},$ $1\le p\le \infty,$ $\beta\in\mathbb{R},$ with high exponents of smoothness $r\ (r-1\ge \sqrt{n}).$  We also obtain similar estimates for functional classes $W^r_{\beta,1}$ in metrics of the spaces $L_p, 1\le p\le\infty.$

References

A. I. Stepanets, Classification and approximation of periodic functions, Kluwer Acad. Publ., Dordrecht (1995), https://doi.org/10.1007/978-94-011-0115-8 DOI: https://doi.org/10.1007/978-94-011-0115-8

A. I. Stepanets, Methods of approximation theory, VSP, Utrecht (2005), https://doi.org/10.1515/9783110195286 DOI: https://doi.org/10.1515/9783110195286

B. Sz.-Nagy, Uber gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. 1. Periodischer Fall, Ber. Math.-Phys. Kl. Akad. Wiss., Leipzig, 90, 103 – 134 (1938).

S. B. Stečkin, On the best approximation of certain classes of periodic functions by trigonometric polynomials, Izv. Akad. Nauk SSSR, Ser. Mat., 20, 643 – 648 (1956) (in Russian).

V. N. Temlyakov, Approximation of periodic functions, Comput. Math. and Anal. Ser. Nova Sci. Publ. Inc., New York (1993).

A. N. Kolmogorov, On the order of the remainders of the Fourier series of differentiable functions, Selected Works. Mathematics and Mechanics, Nauka, Moscow (1985), p. 179 – 185.

V. T. Pinkevich, On the order of the remainders of the Fourier series of functions differentiable in the sense of Weyl, Izv. Akad. Nauk SSSR, Ser. Mat., 4, 521 – 528 (1940) (in Russian).

S. M. Nikol’skii, An asymptotic estimation of the remainder under approximation by Fourier sums, Dokl. Akad. Nauk SSSR, 32, 386 – 389 (1941) (in Russian).

A. V. Efimov, Approximation of continuous periodic functions by Fourier sums, Izv. Akad. Nauk SSSR, Ser. Mat., 24, 243 – 296 (1960) (in Russian).

S. A. Telyakovskii, On the norms of trigonometric polynomials and approximation of differentiable functions by the linear means of their Fourier series, Tr. Mat. Inst. Akad. Nauk SSSR, 62, 61 – 97 (1961) (in Russian).

S. M. Nikol’skii, Approximation of functions in the mean by trigonometric polynomials, Izv. Akad. Nauk SSSR, Ser. Mat., 10, 207 – 256 (1946) (in Russian).

S. B. Stechkin, S. A. Telyakovskii, On approximation of differentiable functions by trigonometric polynomials in the $L$ metric, Tr. Mat. Inst. Akad. Nauk SSSR, 88, 20 – 29 (1967) (in Russian).

I. G. Sokolov, The remainder term of the Fourier series of differentiable functions, Dokl. Akad. Nauk SSSR, 103, 23 – 26 (1955) (in Russian).

S. G. Selivanova, Approximation by Fourier sums of the functions possessing a derivative satisfying the Lipschitz condition, Dokl. Akad. Nauk SSSR, 105, 909 – 912 (1955) (in Russian).

G. I. Natanson, Approximation by Fourier sums of functions possessing different structural properties on different parts of the domain of definition, Vestn. Leningr. Univ., 19, 20 – 35 (1961) (in Russian).

S. A. Telyakovskii, Approximation of differentiable functions by the partial sums of their Fourier series, Math. Notes, 4, 668 – 673 (1968). DOI: https://doi.org/10.1007/BF01116445

S. A. Telyakovskii, Approximation of functions of high smoothness by Fourier sums, Ukr. Math. J., 41, № 4, 444 – 451 (1989). DOI: https://doi.org/10.1007/BF01060623

S. A. Telyakovskii, On approximation by Fourier sums of differentiable functions of high smoothness, Tr. Mat. Inst. Steklov, 198, 193 – 211 (1992).

S. B. Stechkin, An estimation of the remainders of the Fourier series of differentiable functions, Tr. Mat. Inst. Akad. Nauk SSSR, 145, 126 – 151 (1980) (in Russian).

A. S. Serdyuk, I. V. Sokolenko, Approximation by Fourier sums in classes of differentiable functions with high exponents of smoothness, Methods Funct. Anal. and Top., 25, № 4, 381 – 387 (2019).

A. S. Serdyuk, I. V. Sokolenko, Asymptotic estimates for the best uniform approximations of classes of convolution of periodic functions of high smoothness, J. Math. Sci., 252, 526 – 540 (2021), https://doi.org/10.1007/s10958-020-05178-1 DOI: https://doi.org/10.1007/s10958-020-05178-1

А. С. Сердюк, I. В. Соколенко, Наближення iнтерполяцiйними тригонометричними полiномами в метриках просторiв $L_p$ на класах перiодичних цiлих функцiй, Укр. мат. журн., 71, № 2, 283 – 292 (2019).

A. S. Serdyuk, I. V. Sokolenko, Uniform approximation of classes of $(ψ, {bar β})$-differentiable functions by linear methods, Approx. Theory Funct. and Relat. Probl., 8, № 1, 181 – 189 (2011) (in Ukrainian).

A. S. Serdyuk, I. V. Sokolenko, Approximation by linear methods of classes of $(ψ, {bar β})$-differentiable functions, Approx. Theory Funct. and Relat. Probl., 10, № 1, 245 – 254 (2013) (in Ukrainian).

A. S. Serdyuk, Approximation of classes of analytic functions by Fourier sums in the uniform metric, Ukr. Math. J., 57, № 8, 1275 – 1296 (2005), https://doi.org/10.1007/s11253-005-0261-0 DOI: https://doi.org/10.1007/s11253-005-0261-0

A. S. Serdyuk, Approximation of classes of analytic functions by Fourier sums in the metric of the space $L_p$ , Ukr. Math. J., 57, № 10, 1635 – 1651 (2005)? https://doi.org/10.1007/s11253-006-0018-4 DOI: https://doi.org/10.1007/s11253-006-0018-4

A. S. Serdyuk, Nablizhennya interpolyacijnimi trigonometrichnimi polinomami na klasah periodichnih analitichnih funkcij, Ukr. mat. zhurn., 64, № 5, 698 – 712 (2012).

N. P. Kornejchuk, Tochnye konstanty v teorii priblizheniya, Nauka, Moskva (1987).

A. S. Serdyuk, T. A. Stepanyuk, Nablizhennya klasiv uzagal'nenih integraliv Puassona sumami Fur’є v metrikah prostoriv $L_s$ , Ukr. mat. zhurn., 69, № 5, 695 – 704 (2017).

A. S. Serdyuk, T. A. Stepanyuk, Uniform approximations by fourier sums in classes of generalized Poisson integrals, Anal. Math., 45, № 1, 201 – 236 (2019), https://doi.org/10.1007/s10476-018-0310-1 DOI: https://doi.org/10.1007/s10476-018-0310-1

E. T. Uitteker, Dzh. N. Vatson, Kurs sovremennogo analiza, Fizmatgiz, Moskva (1983).

Published
17.06.2022
How to Cite
Serdyuk A. S., and Sokolenko I. V. “Approximation by Fourier Sums in Classes of Weyl – Nagy Differentiable Functions With High Exponent of Smoothness”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 5, June 2022, pp. 685 -00, doi:10.37863/umzh.v74i5.7136.
Section
Research articles