Majorization in de Branges spaces I. Representability of subspaces

TL;DR

This paper investigates which subspaces of de Branges spaces can be characterized through majorization on specific subsets of the upper half-plane, revealing different behaviors depending on the set's proximity to the real axis or infinity.

Contribution

It provides a detailed analysis of the conditions under which subspaces of de Branges spaces are representable via majorization, depending on the chosen subset D.

Findings

Representation depends on the location of D in the upper half-plane.

Different behaviors occur when D is near the real axis versus near infinity.

Results clarify the structure of subspaces generated by majorization.

Abstract

In this series of papers we study subspaces of de Branges spaces of entire functions which are generated by majorization on subsets $D$ of the closed upper half-plane. The present, first, part is addressed to the question which subspaces of a given de Branges space can be represented by means of majorization. Results depend on the set $D$ where majorization is permitted. Significantly different situations are encountered when $D$ is close to the real axis or accumulates to $i\infty$.