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On the best polynomial approximation of entire transcendental functions in banach spaces. I

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Abstract

We study the behavior of the best approximationsE n (ϕ) p of entire transcendental functions ϕ(z) of the order ρ=∞ by polynomials of at mostn th degree in the metric of the Banach space E′p(Ω) of functions /tf(z) analytic in a bounded simply connected domain Ω with rectifiable Jordan boundary and such that

$$\left\| f \right\|_{E'_p } = \left\{ {\iint_\Omega {\left| {f\left( z \right)} \right|^p }dxdy} \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}}< \infty $$

.

In particular, we describe the relationship between the best approximationsE n (ϕ)p and theq-order andq-type of the function ϕ(z).

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Authors and Affiliations

  1. Institute of Geotechnical Mechanics, Ukrainian Academy of Sciences, Dnepropetrovsk

    S. B. Vakarchuk

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  1. S. B. Vakarchuk
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1123–1133, September, 1994.

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Vakarchuk, S.B. On the best polynomial approximation of entire transcendental functions in banach spaces. I. Ukr Math J 46, 1235–1247 (1994). https://doi.org/10.1007/BF01059415

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  • DOI: https://doi.org/10.1007/BF01059415

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