On the approximation of analytic functions represented by Dirichlet series (Q1101843)

Let \(\pi_ n\) denote the class of exponential polynomials \(\sum^{n}_{k=1}c_ k \exp (s\lambda_ k)\) of degree at most n. Functions in these classes are used to approximate a given function \(f(s)=\sum^{\infty}_{k=1}a_ k \exp (s\lambda_ k)\) analytic in the half-plane Re s\(<\alpha\) \((-\infty <\alpha <\infty)\) with error \[ E_ n(f,\beta)=\inf_{p\in \pi_ n}\sup_{-\infty \leq t<\infty}| f(\beta +it)-p(\beta +it)| \quad, \] for \(n\in {\mathbb{N}}\), Re \(s\leq \beta <\alpha\). Previously known results of \textit{A. Nautiyal} and \textit{D. P. Shukla} [Indian. J. Pure Appl. Math. 14, 722-727 (1983; Zbl 0522.30031)] are extended and give characterizations of lower order and lower type of f(s) in terms of \(E_ n(f,\beta)\).