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On inclusion relations between weighted spaces of entire functions

Gerhard Schindl

Abstract

We characterize the inclusions of weighted classes of entire functions in terms of the defining weights resp. weight systems. First we treat weights defined in terms of a so-called associated weight function where the weight(system) is based on a given sequence. The abstract weight function case is then reduced to the weight sequence setting by using the so-called associated weight sequence. As an application of the main statements we characterize closedness under point-wise multiplication of these classes.

On inclusion relations between weighted spaces of entire functions

Abstract

We characterize the inclusions of weighted classes of entire functions in terms of the defining weights resp. weight systems. First we treat weights defined in terms of a so-called associated weight function where the weight(system) is based on a given sequence. The abstract weight function case is then reduced to the weight sequence setting by using the so-called associated weight sequence. As an application of the main statements we characterize closedness under point-wise multiplication of these classes.
Paper Structure (22 sections, 41 theorems, 161 equations)

This paper contains 22 sections, 41 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Weights, weighted spaces and growth conditions
  3. Basic convention and comments
  4. Weight systems and corresponding weighted classes
  5. Weight sequences and conditions
  6. Associated weight function
  7. Weighted spaces of entire functions defined in terms of $\omega_M$
  8. On the class of (log-)convex weights $v$
  9. From weight functions to weight sequences
  10. Technical growth conditions on weight functions
  11. Exponential-type weight systems in the literature
  12. Characterization of inclusions and equalities of weighted classes
  13. Characterization for essential weights
  14. The dilatation-type weight sequence setting
  15. The exponential-type weight sequence setting
  16. ...and 7 more sections

Key Result

Proposition 2.2

Let $v$ and $w$ be (normalized) weight functions. If $vompreceqc w$, then and if $vompreceqpowc w$, then with continuous inclusions.

Theorems & Definitions (93)

  • Definition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • proof
  • Lemma 2.9
  • ...and 83 more