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Generalized growth and approximation of entire function solution of Helmholtz equation in Banach spaces. (English) Zbl 1341.30026

Summary: In this paper, we study the generalized growth and polynomial approximation of entire function solution of Helmholtz equation in \(R^2\) in Smirnov spaces [\(\varepsilon_p (S)\) and \(\varepsilon^\prime_p(S)\), \(1\leq p\leq \infty \)] where \(S\) is finitely simply connected domain in the complex plane with the boundary that belongs to the Al’per class [S. Ya. Al’per, Izv. Akad. Nauk SSSR, Ser. Mat. 19, No. 6, 423–444 (1955; Zbl 0065.30402)]. Some bounds on generalized order and generalized type of entire solution of Helmholtz equation have been obtained in terms of the coefficients and approximation errors using function theoretic methods. Our results extend and improve the results of D. Kumar [J. Appl. Anal. 18, No. 2, 179–196 (2012; Zbl 1276.30045)].

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
41A10 Approximation by polynomials
30H99 Spaces and algebras of analytic functions of one complex variable
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