On extremal problems of approximation theory for classes of functions of two discrete variables in $$$l^2_{q,p}$$$

S.B. Vakarchuk; M.B. Vakarchuk

doi:10.15421/240709

On extremal problems of approximation theory for classes of functions of two discrete variables in $$$l^2_{q,p}$$$

S.B. Vakarchuk (Academy of Customs Service of Ukraine), https://orcid.org/0000-0002-2562-8844
M.B. Vakarchuk (Dnipropetrovsk National University), https://orcid.org/0000-0001-8606-9057

Abstract

For classes of $$$2\pi$$$-periodic functions of two discrete variables, defined on grid set $$$\sigma_{q,p}$$$, we found exact values of Kolmogorov and linear quasiwidths of classes $$$W^{r,p}(l^2_{q,p})$$$ in $$$l^2_{q,p}$$$ space.

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References

Babenko A.G. Extremal properties of polynomials and exact estimates of mean-square approximations: Diss. for cand.phys.-math.sc., Sverdlovsk, 1987. (in Russian)

Babenko V.F., Kofanov V.A., Pichugov S.A. "Approximation of functions of discrete argument", Optimization of approximation methods, Kyiv, 1992; pp. 14-26. (in Russian)

Vakarchuk S.B. "On approximation of functions of two variables by generalized polynomials", Trudy MIAN SSSR. Function theory and related problems of analysis, 1987; 180: pp. 77-78. (in Russian)

Vakarchuk S.B. "On exact values of quasiwidths of some classes of periodic functions of two variables", Approx. theory and related prob. of analysis and topology. Sb. nauch. tr. In-ta matem. AN USSR, Kyiv, 1987; pp. 15-20. (in Russian)

Vakarchuk S.B., Vakarchuk M.B. "On exact values of widths of classes of functions of discrete argument in $$$l^2_q$$$", Res. Math., 2000; 8: pp. 35-39. (in Russian)

Velikina Yu.V., Velikin V.L. "Widths of sets of functions of discrete argument", Ukrainian Math. J., 1996; 48(10): pp. 1311-1320. (in Russian) doi:10.1007/BF02377817

Grigorian A.L. "Discrete Lebesgue constants", Matem. zametki, 1983; 34(6): pp. 857-866. (in Russian) doi:10.1007/BF01157408

Guter R.S., Kudriavtsev L.D., Levitan B.M. Elements of function theory. Functions of real variable. Approximation of functions. Almost-periodic functions, 1963. (in Russian)

Demiakov V.F., Malozemov V.N. Introduction to minimax, 1972. (in Russian)

Zygmund A. Trigonometric series. Vol. 2, 1965. (in Russian)

Kalashnikov M.D. "On polynomials of the best (square) approximation on given point set", Dokl. AN SSSR, 1955; 105(4): pp. 634-636. (in Russian)

Lusternik L.A., Sobolev V.I. Elements of functional analysis, 1965. (in Russian)

Potapov M.K. "Research of some classes of functions by means of 'angle' approximation", Trudy MIAN SSSR im. V.A. Steklova, 1972; 117: pp. 256-291. (in Russian)

Potapov M.K. "Approximation by 'angle' and embedding theorems", Math. Balcanica, 1972; 2: pp. 183-198. (in Russian)

Temlyakov V.N. "Approximation of functions with bounded mixed derivative", Trudy MIAN SSSR, 1986; 178: pp. 3-112. (in Russian)

Sharapudinov I.I. "Approximation of discrete functions and Chebyshev polynomials that are orthogonal on uniform grid", Matem. zametki, 2000; 67(3): pp. 460-470. (in Russian) doi:10.1007/BF02676675

Cheney E.W. Introduction to Approximation Theory, McGraw-Hill Book Co., 1966.

Rivlin T.J. An Introduction to the Approximation of Functions, Blaisdell Pub. Co., 1969.