Homogeneous spaces of entire functions

TL;DR

This paper provides a comprehensive analysis of homogeneous spaces of entire functions, explicitly describing their structure, correcting past errors, and offering detailed proofs of known and new results in the theory.

Contribution

It offers a complete, detailed account of homogeneous spaces, including explicit proofs, corrections of past mistakes, and descriptions of their structure and associated measures.

Findings

Explicit formulas for homogeneous spaces and their measures

Correction of a previously unnoticed mistake in the theory

Detailed proofs of the relations between the objects involved

Abstract

Homogeneous spaces are de Branges' Hilbert spaces of entire functions with the property that certain weighted rescaling transforms induce isometries of the space into itself. A classical example of a homogeneous space is the Paley-Wiener space of entire functions with exponential type at most a being square integrable on the real axis. Other examples occur in the theory of the Bessel equation. Being homogeneous is a strong property, and one can describe all homogeneous spaces, their structure Hamiltonians, and the measures associated with chains of such spaces, explicitly in terms of powers, logarithms, and confluent hypergeometric functions. The theory of homogeneous spaces was in large parts settled by L.de Branges in the early 1960's. However, in his work some connections and explicit formulae are not given, some results are stated without a proof, and last but not least a mistake occurs which seemingly remained unnoticed up to the day. In this paper we give a detailed account on homogeneous spaces. We provide explicit proofs for all formulae and relations between the mentioned objects, and correct the mentioned mistake.