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

#### 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

$$\sup\limits_{\substack{f\in L^{(r)}\\f \ne const}} \frac{\sigma^r A_{\sigma}(f)_L}{\omega (f^{(r)}, \pi/\sigma)_L} = \frac{K_L}{2}$$

$$A_{\sigma}(W^r H^{\omega}_0)_C = \frac{1}{\sigma^{r+1}} \int\limits_0^{\pi} \Phi_{\pi, r}(t)\omega'(t/\sigma)dt$$

for $$$\sigma > 0$$$, $$$r = 1, 2, 3, \ldots$$$ and concave modulus of continuity.

Also, we calculate the supremum

$$\sup\limits_{\substack{f\in L^{(r)}\\f \ne const}} \frac{\sigma^r A_{\sigma}(f)_L}{\omega (f^{(r)}, \pi/\sigma)_L} = \frac{K_L}{2}$$

#### Full Text:

PDF (Русский)#### 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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