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On the best polynomial approximation of entire transcendental functions of generalized order

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Ukrainian Mathematical Journal Aims and scope

We prove a Hadamard-type theorem that associates the generalized order of growth \(\rho_{f}^*({\alpha}, {\beta})\) of an entire transcendental function ƒ with the coefficients of its expansion in a Faber series. This theorem is an extension of one result of Balashov to the case of a finite simply connected domain G with boundary γ belonging to the Al'per class Λ*. Using this theorem, we obtain limit equalities that associate \(\rho_{f}^*({\alpha}, {\beta})\) with a sequence of the best polynomial approximations of ƒ in certain Banach spaces of functions analytic in G.

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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1011–1026, August, 2008.

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Vakarchuk, S.B., Zhir, S.I. On the best polynomial approximation of entire transcendental functions of generalized order. Ukr Math J 60, 1183–1199 (2008). https://doi.org/10.1007/s11253-009-0134-z

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