On the extension of differentiable functions from their monotonicity interval and inequalities of Kolmogorov type

Yu.S. Zagorul'ko; V.A. Kofanov

doi:10.15421/241406

On the extension of differentiable functions from their monotonicity interval and inequalities of Kolmogorov type

Yu.S. Zagorul'ko (Oles Honchar Dnipropetrovsk National University)
V.A. Kofanov (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0003-0392-2257

Abstract

We study the possibility of extension for every function $$$f\in L_{\infty}(\mathbb{R})$$$ from any monotonicity interval $$$I$$$ of function $$$f$$$ to the whole axis with retaining norms of $$$f$$$ and $$$f^{(r)}$$$ on interval.

Keywords

Kolmogorov-type inequalities; the comparison theorem of Kolmogorov

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References

Babenko V.F., Kofanov V.A., Pichugov S.A. "Exact constants of Kolmogorov type with bounded higher derivative in case of small smoothness", Ukrainian Math. J., 2001; 53(10): pp. 1298-1308. (in Russian) doi:10.1023/A:1015226223806

Babenko V.F., Korneichuk N.P., Kofanov V.A., Pichugov S.A. Inequalities for derivatives and their applications, Naukova dumka, Kyiv, 2003; 590 p. (in Russian)

Kolmogorov A.N. "On inequalities between upper bounds of consecutive derivatives of the function on infinite interval", Izbr. tr. Matematika, mekhanika, Nauka, Moscow, 1985; pp. 252-263. (in Russian)