The Stieltjes moment problem in Gelfand-Shilov spaces defined by weight sequences in the absence of derivation closedness

TL;DR

This paper investigates the Stieltjes moment problem within Gelfand-Shilov spaces defined by weight sequences, extending the target space for the moment mapping and analyzing injectivity and surjectivity under weaker conditions than derivation closedness.

Contribution

It introduces a new framework for the Stieltjes moment problem in Gelfand-Shilov spaces, allowing larger target spaces and relaxing conditions on weight sequences, with detailed analysis of the moment mapping's properties.

Findings

Characterization of injectivity and surjectivity of the moment mapping.

Extension of the target space for the moment mapping.

Results for weight sequences with fast and regular growth.

Abstract

The Stieltjes moment problem is studied in a new framework within the general Gelfand-Shilov spaces defined via weight sequences. The novelty consists of allowing for a naturally larger target space for the moment mapping, which sends a function to its sequence of Stieltjes moments. The motivation comes from a recent version of the Borel-Ritt theorem, concerning the surjectivity of the Borel mapping in Carleman-Roumieu ultraholomorphic classes in sectors, whose defining weight sequence is subject to the condition, weaker than derivation closedness, of having shifted moments. The injectivity and surjectivity of the moment mapping in this new setting is studied and, in some cases, characterized. Finally, results are provided for general weight sequences of fast and regular enough growth when the condition of shifted moments fails to hold.