Some sharp inequalities for approximations of periodic functions in $$$L_1$$$ space

V.F. Babenko

doi:10.15421/248701

Some sharp inequalities for approximations of periodic functions in $$$L_1$$$ space

V.F. Babenko (Dnipropetrovsk State University), https://orcid.org/0000-0001-6677-1914

Abstract

We provide sharp estimates of Jackson's inequalities type for the best $$$(\alpha, \beta)$$$-approximations in the space $$$L_1$$$ of periodic functions that are representable as the convolution of the kernel $$$K$$$ that does not increase the number of sign alternations with functions $$$\varphi \in C$$$, by means of convolutions of the kernel $$$K$$$ with the functions that are piecewise-constant in the intervals $$$\bigl( \frac{l \pi}{n}, \frac{(l+1)\pi}{n} \bigr)$$$.

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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.

Babenko K.I. Lectures on Approximation Theory, 1970. (in Russian)

Babenko V.F. "Non-symmetric approximations in spaces of integrable functions", Ukrainian Math. J., 1982; 34(4); pp. 409-416. (in Russian) doi:10.1007/BF01091584

Korneichuk N.P. Splines in approximation theory, Nauka, 1984. (in Russian)

Babenko V.F. "Extremal problems of approximation theory and inequalities for rearrangements", Dokl. AN SSSR, 1986; 290(5). (in Russian)