Geodesic
Sharp estimates of best approximations in terms of holomorphic functions of Weierstrass-type operators
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 18-60

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We establish the estimates $$ A_\sigma(f)_P\le KP\bigl(\Phi(\mathcal W)f\bigr), $$ where $W$ is a kernel of special type summable on $\mathbb R$, a function $\Phi$ is holomorphic in the neighborhood of the spectrum of $W$, $A_\sigma(f)_P$ is the best approximation of a function $f$ by entire functions of exponential type not greater than $\sigma$, with respect to seminorm $P$. In some cases for the uniform and the integral norm we find the least possible constant $K$. The estimates are obtained by linear methods of approximation. 
@article{ZNSL_2012_404_a1,
     author = {O. L. Vinogradov},
     title = {Sharp estimates of best approximations in terms of holomorphic functions of {Weierstrass-type} operators},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {18--60},
     publisher = {mathdoc},
     volume = {404},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a1/}
}
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O. L. Vinogradov. Sharp estimates of best approximations in terms of holomorphic functions of Weierstrass-type operators. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 27, Tome 404 (2012), pp. 18-60. http://geodesic.mathdoc.fr/item/ZNSL_2012_404_a1/