• DOI:10.1215/S0012-7094-97-08601-4
• Corpus ID: 122725453

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

@article{Hedenmalm1995AHS,
  title={A Hilbert space of Dirichlet series and systems of dilated functions in \$L^2(0,1)\$},
  author={H̊akan Hedenmalm and Peter Lindqvist and Kristian Seip},
  journal={Duke Mathematical Journal},
  year={1995},
  volume={86},
  pages={1-37}
}

• H. Hedenmalm, P. Lindqvist, K. Seip
• Published 1 December 1995
• Mathematics
• Duke Mathematical Journal

For a function $\varphi$ 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 $\varphi(nx)$, $n=1,2,3,\ldots$, 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 $\cal H$ of Dirichlet series $f(s)=\sum_n a_nn^{-s}$, where the coefficients $a_n$ are square summable. It proves useful to model $\cal H$ 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 $\cal H$ and characters $\chi$, $f_\chi(s)=\sum_na_n\chi(n)n^{-s}$ is a vertical limit function of $f$. We study certain probabilistic properties of these vertical limit functions.