Sharp estimates of approximation of classes of differentiable functions by entire functions

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

Abstract


In the paper, we find the sharp estimate of the best approximation, by entire functions of exponential type not greater than $$$\sigma$$$, for functions $$$f(x)$$$ from the class $$$W^r H^{\omega}$$$ such that $$$\lim\limits_{x \rightarrow -\infty} f(x) = \lim\limits_{x \rightarrow \infty} f(x) = 0$$$,
$$A_{\sigma}(W^r H^{\omega}_0)_C = \frac{1}{\sigma^{r+1}} \int\limits_0^{\pi} \Phi_{\pi, r}(t)\omega'(t/\sigma)dt$$
for $$$\sigma > 0$$$, $$$r = 1, 2, 3, \ldots$$$ and concave modulus of continuity.
Also, we calculate the supremum
$$\sup\limits_{\substack{f\in L^{(r)}\\f \ne const}} \frac{\sigma^r A_{\sigma}(f)_L}{\omega (f^{(r)}, \pi/\sigma)_L} = \frac{K_L}{2}$$

References


Akhiezer N.I. Lectures on Approximation Theory, Nauka, 1965. (in Russian)

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

Dzyadyk V.K. "On supremums of the best approximations on some classes of continuous functions, defined on real domain", DAN URSR, Ser. A, 1975; 7. (in Ukrainian)

Gromov A.Yu. "Exact constant in Jackson's theorem on the best approximation, by entire functions of exponential type", Res. Math. 1975; pp. 210-211. (in Russian)

Vladimirov V.S. Generalized functions in mathematical physics, 1976. (in Russian)

Gromov A.Yu. "On sharp constants of approximation, by entire functions, for differentiable functions", Res. Math. 1976; pp. 17-21. (in Russian)




DOI: https://doi.org/10.15421/247701

  

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Copyright (c) 1977 V.F. Babenko, A.Yu. Gromov

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