A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$

Abstract

For a function $\phi$ in $L^2(0,1)$, extended to the whole real line as an odd periodic function of period 2, we ask when the collection of dilates $\phi (nx)$, $n=1,2,3,\dots$, constitutes a Riesz basis or a complete sequence in $L^2(0,1)$. The problem translates into a question concerning multipliers and cyclic vectors in the Hilbert space $\mathcal{H}$ of Dirichlet series $f(s)={\sum }_n{a}_n{n}^{-s}$, where the coefficients $a_n$ are square summable. It proves useful to model $\mathcal{H}$ as the $H^2$ space of the infinite-dimensional polydisk, or, which is the same, the $H^2$ space of the character space, where a character is a multiplicative homomorphism from the positive integers to the unit circle. For given $f$ in $\mathcal{H}$ and characters $\chi$, $f_{\chi }(s)={\sum }_n{a}_n\chi (n)n^{-s}$ is a vertical limit function of $f$. We study certain probabilistic properties of these vertical limit functions.