Beurling primes with large oscillation

Beurling Zeta Functions, Generalised Primes, and Fractal Membranes

We study generalised prime systems P (1 < p1 ≤ p2 ≤ · · · , with pj ∈ R tending to infinity) and the associated Beurling zeta function ζP(s) = ∏∞ j=1(1 − p−s j )−1. Under appropriate assumptions, we establish various analytic properties of ζP(s), including its analytic continuation and we characterise the existence of a suitable generalised functional equation. In particular, we examine the rel...

Weak Beurling Property and Extensions to Invertibility

Let E , E∗ be Hilbert spaces. Given a Hilbert space H of holomorphic functions in a domain Ω in C, consider the multiplier space MH(E , E∗). It is shown that for “nice enough” H, the following statements are equivalent for f ∈ MH(E , E∗): (1) There exists a g ∈ MH(E∗, E) such that g(z)f(z) = IE for all z ∈ Ω. (2) There exists a Hilbert space Ec and fc ∈ MH(Ec, E∗) such that F (z) := [ f(z) fc(z...

Constructive Decomposition of Functions of Finite Central Mean Oscillation

The space CMO of functions of finite central mean oscillation is an analogue of BMO where the condition that the sharp maximal function is bounded is replaced by the convergence of the sharp function at the origin. In this paper it is shown that each element of CMO is a singular integral image of an element of the Beurling space B2 of functions whose Hardy-Littlewood maximal function converges ...

Oscillation Theorems for Primes in Arithmetic Progressions and for Sifting Functions

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by S. Li and H. Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for dis...