On one extremal property of Korovkin's means

V.F. Babenko (Dnipropetrovsk State University), https://orcid.org/0000-0001-6677-1914
S.A. Pichugov (Dnipropetrovsk State University), https://orcid.org/0000-0002-4263-4429

Abstract


We point out that
$$\inf\limits_{L \in L_n} \sup\limits_{\substack{f \in C_{2\pi}\\f \ne const}} \frac{\max \| f(x) - L(f, x) \|}{\omega^*_2(f, \pi/n + 1)} = \frac{1}{2}$$
where $$$C_{2\pi}$$$ is the space of periodic continuous functions on real domain, $$$L_n$$$ is the set of linear operators that map $$$C_{2\pi}$$$ to the set of trigonometric polynomials of order no greater than $$$n$$$ ($$$n = 0,1,\ldots$$$), $$$\omega_2(f, t) = \sup\limits_{x, |h| \leqslant t} |f(x-h) - 2f(x) + f(x+h)|$$$, $$$\omega^*_2(f, t)$$$ is the concave hull of the function $$$\omega_2(f, t)$$$. In this equality, the infimum is attained for Korovkin's means.

References


Davidchik A.N., Ligun A.A. "To Jackson's theorem", Matem. zametki, 1974; 16(6): pp. 681-690. (in Russian) doi:10.1007/BF01149787

Korovkin P.P. Linear operators and approximation theory, Fizmatgiz, 1959. (in Russian)




DOI: https://doi.org/10.15421/247702

  

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Copyright (c) 1977 V.F. Babenko, S.A. Pichugov

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