On widths of one class of periodic functions

V.G. Doronin (Dnipropetrovsk State University)
A.A. Ligun (Dnipropetrovsk State University)

Abstract


In the paper, we have found the A.N. Kolmogorov's width of the class $$$W^r L^+_p$$$ ($$$r=1,2,\ldots$$$, $$$1 \leqslant p \leqslant \infty$$$) of all $$$2\pi$$$-periodic functions $$$f(x)$$$ whose $$$(r-1)$$$-th derivative $$$f^{(r-1)}(x)$$$ is absolutely continuous and $$$\| f^{(r)}_+ \|_p \leqslant 1$$$.

References


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)

Doronin V.G., Ligun A.A. "Exact values of upper bounds of the best approximations of $$$W^r_+ L_{\Phi}$$$ classes in $$$L$$$ metric", Res. Math., 1976; pp. 25-34. (in Russian)

Motornyi V.P., Ruban V.I. "Widths of some classes of differentiable periodic functions in $$$L$$$ space", Matem. zametki, 1975; 17(4): pp. 531-543. (in Russian) doi:10.1007/BF01105381

Korneichuk N.P. Extremum problems in approximation theory, 1976. (in Russian)

Stein E.M. "Functions of exponential type", Ann. Math., 1957; 65(3): pp. 582-592. doi:10.2307/1970066




DOI: https://doi.org/10.15421/247704

  

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Copyright (c) 1977 V.G. Doronin, A.A. Ligun

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ISSN (Online): 2664-5009
ISSN (Print): 2664-4991
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