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On approximation of analytic functions by periodic Hurwitz zeta-functions

    Darius Siaučiūnas Affiliation
    ; Violeta Franckevič Affiliation
    ; Antanas Laurinčikas Affiliation

Abstract

The periodic Hurwitz zeta-function ζ(s, α; a), s = σ +it, with parameter 0 < α ≤ 1 and periodic sequence of complex numbers a = {am } is defined, for σ > 1, by series sum from m=0 to ∞ am / (m+α)s, and can be continued moromorphically to the whole complex plane. It is known that the function ζ(s, α; a) with transcendental or rational α is universal, i.e., its shifts ζ(s + iτ, α; a) approximate all analytic functions defined in the strip D = { s ∈ C : 1/2 < σ < 1. In the paper, it is proved that, for all 0 < α ≤ 1 and a, there exists a non-empty closed set Fα,a of analytic functions on D such that every function f ∈ Fα,a can be approximated by shifts ζ(s + iτ, α; a).


First Published Online: 21 Nov 2018

Keyword : Hurwitz zeta-function, periodic Hurwitz zeta-function, universality, weak con- vergence of probability measures

How to Cite
Siaučiūnas, D., Franckevič, V., & Laurinčikas, A. (2019). On approximation of analytic functions by periodic Hurwitz zeta-functions. Mathematical Modelling and Analysis, 24(1), 20-33. https://doi.org/10.3846/mma.2019.002
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Jan 1, 2019
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