On relative widths of classes of multivariable functions that are defined as a convolution

L. Azar

doi:10.15421/249801

On relative widths of classes of multivariable functions that are defined as a convolution

L. Azar (Dnipropetrovsk State University)

Abstract

We find exact order of estimate of relative widths in $$$L_1$$$ space of classes of periodic multivariable functions that are defined as a convolution of the product of one-dimensional kernels with functions from unit ball of $$$L_1$$$ space. By this we generalize known results on relative widths of classes of functions that are defined by bounds for $$$L_1$$$-norms of mixed derivatives.

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References

Babenko V.F. "On the best uniform approximations by splines under the constraints on their derivatives", Matem. zametki, 1991; 50(6): pp. 24-30. (in Russian) doi:10.1007/BF01158262

Babenko V.F. "On the best $$$L_1$$$-approximations by splines under the constraints on their derivatives", Matem. zametki, 1992; 51(5): pp. 12-19. (in Russian) doi:10.1007/BF01262172

Babenko V.F. "Mean approximations under constraints for derivatives of approximating functions", Vopr. analiza i priblizh., 1989; pp. 9-18. (in Russian)

Babenko V.F. "On the relative widths of classes of functions with bounded mixed derivative", East J. Approx., 1996; 2(3).

Babenko V.F., Azar L.E. "On relative widths of classes of multivariable functions that are defined by fractional derivatives", Sborn. nauch. trudov. Dnepropetr., 1998; pp. 96-103. (in Russian)

Babenko K.I. "On approximation of periodic multivariable functions by trigonometric polynomials", Dokl. AN SSSR, 1960; 132(2): pp. 247-250. (in Russian)

Dzyadyk V.K. Introduction to the theory of uniform approximation of functions by polynomials, 1977. (in Russian)

Kolmogorov A.N. "On the best approximation of functions from given class of functions", Matematika i mekhanika. Izbr. trudy, Nauka, Moscow, 1985; pp. 186-189. (in Russian)

Konovalov V.N. "Estimates of widths of Kolmogorov type for classes of differentiable functions", Matem. zametki, 1984; 35(3): pp. 369-380. (in Russian) doi:10.1007/BF01139916

Korneichuk N.P. Exact constants in approximation theory, Nauka, 1987. (in Russian)

Mitiagin B.S. "Approximation of functions in $$$L_p$$$ and $$$C$$$ spaces on torus", Matem. sbornik, 1962; 58(3): pp. 397-414. (in Russian)

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

Tikhomirov V.M. Some problems of approximation theory, Izd-vo Moskovskogo un-ta, 1976. (in Russian)