Approximation of an Analytic Function Represented by Vector Valued Generalized Dirichlet Series

Abstract

Andre Boivin and Changzhong Zhu  introduced the Dirichlet series with complex exponents and obtained the growth properties of entire functions represented by these series. Later, in 2009, Wen Ping Huang, Ju Hong Ning and Jin Tu  made independent studies on these series. In our earlier work, we have introduced the concept of growth of analytic functions represented by vector valued Dirichlet  series  with complex exponents. In these series, we have taken the coefficients from a complex Banach  algebra. In the present paper, we have introduced the approximation error of these series with respect to a class of exponential polynomials. We have characterized the order and the type  of the analytic function f(s) represented by a vector valued Dirichlet series with complex exponents in terms of the rate of decay of the approximation error introduced. Our results generalize some of the earlier results obtained by A.Nautiyal and D.P.Shukla for classical Dirichlet series.