On extremal subspaces for widths of classes of convolutions

N.V. Parfinovich

doi:10.15421/241708

On extremal subspaces for widths of classes of convolutions

N.V. Parfinovich (Oles Honchar Dnipropetrovsk National University)

Abstract

We obtained the exact values of the best $$$L_1$$$-approximations of the classes $$$K*F$$$ ($$$r\in \mathbb{N}$$$) of periodic functions $$$K*f$$$ such that $$$f$$$ belongs to a given rearrangement-invariant set $$$F$$$ and $$$K$$$ is $$$2\pi$$$-periodic, not increasing oscillation, kernel, by subspaces of generalized polynomial splines with nodes at points $$$2k\pi / n$$$ ($$$n\in \mathbb{N}$$$, $$$k\in \mathbb{Z}$$$). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

Keywords

best approximation; width; periodic function; convolution; spline

Full Text:

PDF (Українська)

References

Mairhuber J.C., Schoenberg I.J., Williamson R.E. "On variation diminishing transformations on the circle", Rend. Circ. mat. Polermo., 1959; 8(2): pp. 241-270. doi:10.1007/BF02843691

Karlin S. Total Positivity, Stanford Univ. Press, Stanford, California, 1968; Vol. 1: 540 p.

Korneichuk N.P. Extremum problems in approximation theory, Moscow, 1976; 320 p. (in Russian)

Korneichuk N.P., Ligun A.A., Doronin V.G. Approximation with constraints, Kyiv, 1982; 250 p. (in Ukrainian)

Korneichuk N.P. Exact constants in approximation theory, Nauka, Moscow, 1987; 424 p. (in Russian)

Nikolsky S.M. "Mean approximation of functions by trigonometrical polynomials", Izv. AN SSSR. Ser.: Matematika, 1946; 10(3): pp. 207-256. (in Russian)

Taikov L.V. "On mean approximation of certain classes of analytical functions", Trudy Matem. in-ta AN USSR, 1967; 88: pp. 61-70. (in Russian)

Turovets S.P. "On the best mean approximation of differentiable functions", Doklady AN Ukrainian SSR. Ser. A, 1968; 5: pp. 417-421. (in Russian)

Ligun A.A. "Inequalities for upper bounds of functionals", Analysis Math., 1976; 2(1): pp. 11-40. doi:10.1007/BF02079905

Makovoz Yu.I. "Widths of certain functional classes in $$$L$$$ space", Izv. AN BSSR. Ser. fiz.-matem., 1969; 4: pp. 19-28. (in Russian)

Subbotin Yu.N. "Width of $$$W^r L$$$ class in $$$L(0,2\pi)$$$ and approximation by spline-functions", Matem. zametki, 1970; 7(1): pp. 43-52. (in Russian) doi:10.1007/BF01093337

Subbotin Yu.N. "Approximation by spline-functions and estimates of widths", Trudy MIAN USSR, 1971; 109: pp. 35-60. (in Russian)

Makovoz Yu.I. "On one method of obtaining lower bounds for set widths in Banach spaces", Matem. zametki, 1972; 87(1): pp. 136-142. (in Russian) doi:10.1070/SM1972v016n01ABEH001354

Pinkus A. "On n-width of periodic functions", J. Anal. Math, 1979; 35: pp. 209-235.

Babenko V.F. "Approximations, widths and optimal quadrature formulae for classes of periodic functions with rearrangement invariant sets of derivatives", Analysis Mathematica, 1987; 13: pp. 281-306. doi:10.1007/BF01909434

Babenko V.F. "Approximation of classes of convolutions", Siberian Math. J., 1987; 28(5): pp. 6-21. (in Russian) doi:10.1007/BF00969308

Pinkus A. n-Widths in approximation theory, Springer-Verlag, Berlin, 1985; 290 p.

Babenko V.F. Extremal problems of approximation theory and non-symmetric norms: Diss. Doc. Phys.-Math. Sc., Dnipropetrovsk, DSU, 1987; 275 p. (in Russian)

Babenko V.F., Parfinovich N.V. "Exact values of the best approximations of classes of periodic functions by splines of defect 2", Matem. zametki, 2009; 85(4): pp. 538-551. (in Russian) doi:10.1134/S0001434609030237

Kolmogorov A.N. "On inequalities between upper bounds of consecutive derivatives of the function on infinite interval", Uch. zap. MGU. Matematika, 1939; 30(3): pp. 3-16. (in Russian)

Korneichuk N.P., Babenko V.F., Ligun A.A. Extremum properties of polynomials and splines, Naukova dumka, Kyiv, 1992; 304 p. (in Russian)