A HILBERT SPACE OF DIRICHLET SERIES AND SYSTEMS OF DILATED FUNCTIONS IN L^{2}(0, 1)

HÅKAN HEDENMALM PETER LINDQVIST KRISTIAN SEIP

Abstract. For a function φ 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 φ(nx), n = 1, 2, 3, …, 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 H of Dirichlet series f(s) = ∑_{n} a_{n}n^{−s}, where the coefficients a_{n} are square summable. It proves useful to model H as the H^{2} space of the infinitedimensional 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 H and characters χ, f_{χ}(s) = ∑_{n} a_{n}χ(n)n^{−s} is a vertical limit function of f. We study certain probabilistic properties of these vertical limit functions.