Abstract
We study the order (velocity) of the approximation of functions on the axis by entire functions of exponential type not higher than σ as σ → ∞ (the linear and best approximations). The exact order of approximation of individual functions on ℝd by the classical summation methods of Fourier integrals (Gauss–Weierstrass, Bochner–Riesz, Marcinkiewicz) and the nonclassical Bernstein–Stechkin method is found. For functions on a torus, similar theorems of approximation by polynomials were obtained previously.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. N. Bernstein, “On properties of homogeneous functional classes,” Dokl. Akad. Nauk SSSR, Ser. Mat., 47, No. 2, 111–114 (1947).
B. R. Draganov, “Exact estimates of the rate of approximation of convolution operators,” J. Appr. Theory, 162, 952–979 (2010).
O. V. Kotova and R. M. Trigub, “Approximative properties of the summation of Fourier integrals,” Dopov. NAN Ukr., No. 1, 13–19 (2015).
O. V. Kotova and R. M. Trigub, “A new sufficient condition of membership to the algebra of absolutely convergent Fourier integrals and its application to the questions of summability of double Fourier series,” Ukr. Mat. Zh., 67, No. 8 (2015).
O. I. Kuznetsova and R. M. Trigub, “Two-sided estimates of approximation of functions by Riesz and Marcinkiewicz means,” Dokl. Akad. Nauk SSSR, 251, No. 1, 34–36 (1980).
V. V. Lebedev and A. M. Olevskii, “Fourier Lp-multopliers with bounded powers,” Izv. Ross. Akad. Nauk, Ser. Mat., 70, No. 3, 129–166 (2006).
B. Ya. Levin, Lectures on Entire Functions, Amer. Math. Soc., Providence, R.I., 1996.
E. Liflyand, S. Samko, and R. Trigub, “The Wiener algebra of absolutely convergent Fourier integrals: an overview,” Anal. and Math. Phys., 2, No. 1, 1–68 (2012).
B. M. Makarov and A. N. Podkorytov, Lectures on Real Analysis [in Russian], BKhV–Petrburg, St.-Petersburg, 2011.
S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems, Springer, Berlin, 1974.
R. K. S. Rathore, “The problem of A. F. Timan on the precise order of decrease of the best approximation,” J. Appr. Theory., 77, 153–166 (1994).
H. S. Shapiro, “Some Tauberian teorems with applications to approximation theory,” Bull. Amer. Math. Soc., 74, 499–504 (1968).
S. B. Stechkin, “On the order of the best approximations of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 15, No. 3, 219–242 (1951).
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.
A. F. Timan, Theory of Approximation of Functions of a Real Variable, Dover, New York, 1964.
R. M. Trigub, “Constructive characteristics of some classes of functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 29, No. 3, 615–630 (1965).
R. M. Trigub, “Linear methods of summation and the absolute convergence of Fourier series,” Izv. Akad. Nauk SSSR, Ser. Mat., 32, No. 1, 24–49 (1968).
R. M. Trigub, “Absolute convergence of Fourier integrals, summability of Fourier series and the approximation of functions on a torus by polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat., 44, No. 6, 1378–1409 (1980).
R. M. Trigub, “The exact order of approximation of periodic functions by Bernstein–Stechkin polynomials,” Mat. Sborn., 204, No. 12, 127–146 (2013).
R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer Acad. Publ., Dordrecht, 2004.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 12, No. 2, pp. 222–242, April–May, 2015.
Rights and permissions
About this article
Cite this article
Kotova, O.V., Trigub, R.M. Approximative properties of the summation methods of Fourier integrals. J Math Sci 211, 668–683 (2015). https://doi.org/10.1007/s10958-015-2623-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-015-2623-y