Joint discrete approximation of analytic functions by shifts of Lerch zeta-functions

• Published in: Mathematical modelling and analysis Vol. 29; no. 2; pp. 178 - 192
• Main Authors: Laurincikas, Antanas, Mikalauskaite, Toma, Siauciunas, Darius
• Format: Journal Article
• Language: English
• Published: Vilnius Vilnius Gediminas Technical University 26.03.2024
• Subjects: Analysis; Analytic functions; Approximation; approximation of analytic functions; Approximation theory; Density; Dirichlet problem; Functions, Zeta; Lerch zeta-functions; Mathematical analysis; Mathematics; Methods; space of analytic functions; weak convergence of probability measures; Lithuania; approximation of analytic functions; weak convergence of probability measures; Lerch zeta-functions; space of analytic functions
• ISSN: 1392-6292; 1648-3510
• DOI: 10.3846/mma.2024.19493

The Lerch zeta-function depends on two real parameters λ and  and, for σ > 1, is defined by the Dirichlet series , and by analytic continuation elsewhere. In the paper, we consider the joint approximation of collections of analytic functions by discrete shifts  with arbitrary 1 and We prove that there exists a non-empty closed set of analytic functions on the critical strip which is approximated by the above shifts. It is proved that the set of shifts approximating a given collection of analytic functions has a positive lower density. The case of positive density also is discussed. A generalization for some compositions is given.