Localization of Zeros in Cauchy–de Branges Spaces

Abstract

We study the class of discrete measures in the complex plain with the following property: up to a finite number, all zeros of any Cauchy transform of the measure (with ℓ 2 -data) are localized near the support of the measure. We find several equivalent forms of this property and prove that the parts of the support attracting zeros of Cauchy transforms are ordered by inclusion modulo finite sets.

Cauchy-de Branges spaces, geometry of their reproducing kernels and multiplication operators

“…Further results related, in particular, to a more subtle property of localization (which is not strong) as well as numerous examples can be found in [1,3].…”

“…E. Abakumov, Yu. Belov and the author in [2,3,4]. These spaces, defined via discrete Cauchy transform and generalizing de Branges spaces, were named the Cauchy-de Branges spaces in [2].…”

“…De Branges and Cauchy-de Branges spaces with "small" measures µ (i.e., with fast decay of µ n ) were studied in [1,3], where the notion of localization was introduced. In what follows we will assume that the sequence T is power separated (see (1.3)).…”

“…It was proved in [1] in the de Brangean setting and in [3] for general CdBspaces that the strong localization property is equivalent to the density of polynomials in L 2 (µ), µ = n µ n δ tn (the elements of L 2 (µ) can be identified with sequences (d n ) such that (d n ) 2 L 2 (µ) = n |d n | 2 µ n < ∞). Therefore, we can add to Theorem 1.9 one more equivalent condition.…”

“…It is easy to show that if M is sufficiently large, e.g., n |t n | M−N < ∞, then g ∈ H(T, A, µ) (see [3,Section 3] for details) and moreover g k → g in H(T, A, µ). Since g k ∈ H 0 , we conclude that g ∈ H 0 .…”