Arithmetical sequences and systems of functional equations

Abstract

This note investigates the arithmetic origin and structure of functional equations of the type $$\frac{1}{n}\sum\limits_{v - 0}^{n - 1} {G(e^{2\pi iv/n} z) = \sum\limits_{d = 1}^\infty {\lambda _n (d)G(z^{nd} )} } $$ with naturaln and complexz, and the closely related $$\frac{1}{n}\sum\limits_{v - 0}^{n - 1} {F\left( {\frac{{x + v}}{n}} \right) = \sum\limits_{d = 1}^\infty {\lambda _n (d)F(dx)} } $$ with realx. We establish some fundamental results concerning their holomorphic, their periodic integrable and their aperiodic continuous solutions, respectively. The main tools are of number theoretic and functional analytic nature.