### 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.

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PDF#### References

Akanksha and G.S.Srivastava ,Growth properties of vector valued Dirichlet series with complex exponents ,(Communicated ).

A.Boivin and Changzhong Zhu, The growth of an entire function and its Dirichlet coefficients and exponents, J. Complex Var. Theory Appl., 48 (5), (2003),397–415.

Wen Ping Huang , Ju Hong Ning and Tu Jin ,Order and type of the generalized Dirichlet series, J. Math. Res. Exposition, 29(6), (2009),1041-1046.

A.Nautiyal and D.P.Shukla, On the approximation of an analytic function by exponential polynomials, Indian J. pure appl. Math.,14 (6), (1983),722-727.

B.L.Srivastava, A study of spaces of certain classes of Vector Valued Dirichlet Series, Thesis, I.I.T Kanpur, (1983).

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