72 research outputs found
Local properties of Hilbert spaces of Dirichlet series
We show that the asymptotic behavior of the partial sums of a sequence of
positive numbers determine the local behavior of the Hilbert space of Dirichlet
series defined using these as weights. This extends results recently obtained
describing the local behavior of Dirichlet series with square summable
coefficients in terms of local integrability, boundary behavior, Carleson
measures and interpolating sequences. As these spaces can be identified with
functions spaces on the infinite-dimensional polydisk, this gives new results
on the Dirichlet and Bergman spaces on the infinite dimensional polydisk, as
well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on
the infinite dimensional unit ball. We use both techniques from the theory of
sampling in Paley-Wiener spaces, and classical results from analytic number
theory.Comment: 27 pages, 1 figur
A quantitative Balian-Low theorem
We study functions generating Gabor Riesz bases on the integer lattice. The
classical Balian-Low theorem restricts the simultaneous time and frequency
localization of such functions. We obtain a quantitative estimate that extends
both this result and other related theorems.Comment: 11 page
Local interpolation in Hilbert spaces of Dirichlet series
We denote by \Hp the Hilbert space of ordinary Dirichlet series with
square-summable coefficients. The main result is that a bounded sequence of
points in the half-plane is an interpolating sequence for \Hp
if and only if it is an interpolating sequence for the Hardy space of the
same half-plane. Similar local results are obtained for Hilbert spaces of
ordinary Dirichlet series that relate to Bergman and Dirichlet spaces of the
half-plane
Modified zeta functions as kernels of integral operators
AbstractThe modified zeta functions ∑n∈Kn−s, where K⊂N, converge absolutely for Res>1. These generalise the Riemann zeta function which is known to have a meromorphic continuation to all of C with a single pole at s=1. Our main result is a characterisation of the modified zeta functions that have pole-like behaviour at this point. This behaviour is defined by considering the modified zeta functions as kernels of certain integral operators on the spaces L2(I) for symmetric and bounded intervals I⊂R. We also consider the special case when the set K⊂N is assumed to have arithmetic structure. In particular, we look at local Lp integrability properties of the modified zeta functions on the abscissa Res=1 for p∈[1,∞]
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