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.
Similar content being viewed by others
References
I. K. Daugavet, Introduction to the Theory of Approximation of Functions [in Russian], Leningrad University, Leningrad (1977).
R. S. Varga, “On an extension of a result of S. N. Bernstein,” J. Approxim. Theory, 1, No. 2, 176–179 (1968).
A. R. Reddy, “Approximation of an entire function,” J. Approxim. Theory, 3, No. 1, 128–137 (1970).
M. N. Sheremeta, “On the relationship between the growth of the maximum of the modulus of an entire function and the moduli of the coefficients of its power expansion,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 100–108 (1967).
M. N. Sheremeta, “On the relationship between the growth of functions of zero order entire or analytic in a disk and the coefficients of their power expansions,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 6, 115–121 (1968).
S. K. Balashov, “On the relationship between the growth of an entire function of generalized order and the coefficients of its power expansion and the distribution of roots,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 8, 10–18 (1972).
S. M. Shah, “Polynomial approximation of an entire function and generalized orders,” J. Approxim. Theory, 19, No 4, 315–32 (1977).
A. R. Reddy, “A contribution to best approximation in the L 2_norm,” J. Approxim. Theory, 11, No. 1, 110–117 (1974).
I. I. Ibragimov and N. I. Shikhaliev, “On the best polynomial approximation in one space of analytic functions,” Dokl. Akad. Nauk SSSR, 227, No. 2, 280–283 (1976).
S. B. Vakarchuk, “On the best polynomial approximation in some Banach spaces of functions analytic in the unit disk,” Mat. Zametki, 55, No. 4, 6–14 (1994).
S. B. Vakarchuk and S. I. Zhir, “On the polynomial approximation of entire transcendental functions, ” Mat. Fiz. Anal., Geom., 9, No. 4, 595–603 (2002).
S. B. Vakarchuk and S. I. Zhir, “On some problems of polynomial approximation of entire transcendental functions,” Ukr. Mat. Zh., 54, No. 9, 1155–1162 (2002).
S. B. Vakarchuk and S. I. Zhir, “On the polynomial approximation of entire transcendental functions in the complex plane,” in: Problems of the Theory of Approximation of Functions and Related Problems, Vol. 2, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2005), pp. 27–42.
A. V. Batyrev, “On the problem of the best polynomial approximation of analytic functions,” Dokl. Akad. Nauk SSSR, 76, No. 2, 173–175 (1951).
I. I. Ibragimov and N. I. Shikhaliev, “On the constructive characteristic of one class of functions of a complex variable,” Dokl. Akad. Nauk SSSR, 236, No. 4, 789–791 (1977).
A. Giroux, “Approximation of entire functions over bounded domains,” J. Approxim. Theory, 28, No. 1, 45–53 (1980).
S. B. Vakarchuk, “On the best polynomial approximation of entire transcendental functions in Banach spaces. I,” Ukr. Mat. Zh., 46, No. 9, 1123–1133 (1994).
S. B. Vakarchuk, “On the best polynomial approximation of entire transcendental functions in Banach spaces. II,” Ukr. Mat. Zh., 46, No. 10, 1318–1322 (1994).
S. Ya. Al'per, “On the uniform approximation of functions of a complex variable in a closed domain, ” Izv. Akad. Nauk SSSR, Ser. Mat., 19, No. 3, 423–444 (1955).
V. I. Smirnov and N. A. Lebedev, Constructive Theory of Functions of Complex Variables [in Russian], Nauka, Moscow (1964).
V. K. Dzyadyk, “On approximation of analytic functions in domains with smooth and piecewise-smooth boundaries,” in: Proceedings of the Third Summer Mathematical School “Constructive Theory of Functions” (Katsiveli, June–July 1965) [in Russian], Naukova Dumka, Kiev (1966), pp. 29–83.
P. K. Suetin, Series in Faber Polynomials [in Russian], Nauka, Moscow (1984).
A. A. Konyushkov, “Best approximations and Fourier coefficients,” Mat. Sb., 44, No. 1, 53–84 (1958).
S. N. Mergelyan, “On some problems of the constructive theory of functions,” Tr. Mat. Inst. Akad. Nauk SSSR, 37, 3–90 (1951).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1011–1026, August, 2008.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-009-0134-z