On widths of one class of periodic functions

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


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 $$$\| f^{(r)}_+ \|_p \leqslant 1$$$.


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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ISSN (Online): 2664-5009
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