Abstract
The paper deals with imbedding theorems and harmonic approximation for classes of periodic functions of several variables. Necessary and sufficient conditions for imbedding are found for various classes of smooth functions. The paper contains asymptotic estimates for fractional order derivatives of multidimensional Dirichlet and Favard kernels, for Fourier best approximation and for the Kolmogorov diameter of smooth function classes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
V. M. Tikhomirov, Some Questions in Approximation Theory [in Russian], Moscow State Univ. (1976).
K. I. Babenko, “On the approximation of a class of periodic functions of several variables by trigonometric polynomials,” Dokl. Akad. Nauk SSSR,132, No. 5, 982–985 (1960).
K. I. Babenko, “On the approximation of periodic functions of several variables by trigonometric polynomials,” Dokl. Akad. Nauk SSSR,132, No. 2, 247–250 (1960).
S. A. Telyakovskii, “Estimates for the derivatives of trigonometric polynomials of several variables,” Sib. Mat. Zh.,4, No. 6, 1404–1441 (1963).
S. A. Telyakovskii, “Some estimates for trigonometric series with quasiconvex coefficients,” Mat. Sb.,63, No. 105, 426–444 (1964).
N. S. Nikol'skaya, “Approximation of differentiable functions of several variables by Fourier sums in the Lp metric,” Sib. Mat. Zh.,15, No. 2, 395–412 (1974).
N. S. Nikol'skaya, “Approximation of periodic functions of a class by Fourier sums,” Sib. Mat. Zh.,16, No. 4, 761–780 (1975).
E. M. Galeev, “Approximation by means of Fourier series of classes of functions with bounded derivatives,” Mat. Zametki,23, No. 2, 197–211 (1978).
V. N. Temlyakov, “Approximation of periodic functions of several variables with bounded mixed difference,” Dokl. Akad. Nauk SSSR,253, No. 3, 544–548 (1980).
V. N. Temlyakov, “Approximation of periodic functions of several variables with bounded mixed difference,” Mat. Sb.,113, No. 1, 65–80 (1980).
Din' Zung, “Some approximation characteristics of classes of smooth functions of several variables in the L2 metric,” Usp. Mat. Nauk,34, No. 4, 189–190 (1979).
Din' Zung and G. G. Magaril-ll'yaev, “Problems of Bernstein and Favard type and mean ε-dimension of some function classes,” Dokl. Akad. Nauk,249, No. 4, 783–786 (1979).
E. M. Galeev, “Order estimates of derivatives of periodic multidimensional Dirichlet α-kernels in the mean norm,” Mat. Sb.,117, No. 1, 32–43 (1982).
S. M. Nikol'skii, “A representation theorem for a class of differentiable functions of several variables by means of entire functions of exponential type,” Dokl. Akad. Nauk SSSR,150, No. 3, 483–487 (1963).
N. S. Bakhvalov, “Imbedding theorems for classes of functions with bounded derivatives,” Vestn. Mosk. Univ. Ser. Mat. Mekh., No. 3, 7–16 (1963).
R. T. Rockafellar, Convex analysis, Princeton Univ. Press (1970).
H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin—Göttingen—Heidelberg (1957).
S. M. Nikol'skii, “An inequality for entire functions and its application to the theory of differentiable functions of several variables,” Tr. Mat. Inst. Steklov,38, 244–278 (1951).
V. M. Tikhomirov, “Some questions in approximation theory,” Dokl. Akad. Nauk SSSR,160, No. 4, 774–777 (1965).
Additional information
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 207–226, 1984.
Rights and permissions
About this article
Cite this article
Zung, D. Approximation of classes of smooth functions of several variables. J Math Sci 35, 2859–2875 (1986). https://doi.org/10.1007/BF01106079
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01106079