Localization of Zeros for Cauchy Transforms

Abstract

Abstract. We study the localization of zeros for Cauchy transforms of discrete measures on the real line. This question is motivated by the theory of canonical systems of differential equations. In particular, we prove that the spaces of Cauchy transforms having the localization property are in one-to-one correspondence with the canonical systems of special type, namely, those whose Hamiltonians consist only of indivisible intervals accumulating on the left. Various aspects of the localization phenomena are studied in details. Connections with the density of polynomials and other topics in analysis are discussed.

Geometry of reproducing kernels in model spaces near the boundary

“…One of our main results says that the overcompleteness is equivalent to the localization property introduced recently by Abakumov, Belov and the first author in [1] in the context of de Branges spaces (i.e., essentially, in the case of model spaces K θ such that σ(θ) consists of one point). Recall that the Stolz angle Γ γ , γ > 1, at the point ζ ∈ T is defined…”

“…In the half-plane setting it is easy to see, using an idea from [1], that in the definition of the localization at ∞ the Stolz angle may be replaced by any domain of the form Γ γ,β = {Im z > γ| Re z| β , |z| > 1} where γ > 0, β ∈ R. Proof. Assume the converse and let f ∈ K Θ have infinitely many zeros in some domain Γ γ,β .…”

Synthesizable differentiation-invariant subspaces

“…Though, at first glance the studied problem is not directly related to the de Branges theory, there exist deep connections between them. Note that this is the case with some other classical problems of harmonic analysis such that description of Fourier frames [15], Beurling-Malliavin type theorems [13], approximation by polynomials [1]. Section 2 is devoted to our toolbox from de Branges theory.…”

“…In this subsection several facts recently discovered by the authors are collected. The next result was used in [1]. Theorem 2.6.…”

Krein-type theorems and ordered structure for Cauchy–de Branges spaces

“…De Branges spaces' theory is a deep and important field which has numerous applications to operator theory, spectral theory of differential operators and even to number theory. For the basics of de Branges theory we refer to de Branges' monograph [7] and to [20]; some further results and applications can be found in [1,4,11,17,18,19].…”