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

Javier Jiménez-Garrido, Ignacio Miguel-Cantero, Javier Sanz, Gerhard Schindl

TL;DR

This work extends the Stieltjes moment problem to Gelfand-Shilov spaces defined by weight sequences, introducing a larger target space $\Lambda_{\{\boldsymbol{M}_{+1}\}}$ when the weaker (sm) condition holds. It develops a Laplace/Borel-type framework linking the moment mapping to asymptotic Borel results, proving injectivity criteria via $\sum_{p} m_p^{-1/2}$ and surjectivity criteria via $\gamma({\boldsymbol{M}})$ (and local/global right inverses with uniform $h$-scaling). When (sm) fails, it introduces a new target sequence $\widetilde{{\boldsymbol{M}}}$ and shows that, under fast, regular growth making $\widetilde{{\boldsymbol{M}}}$ a weight sequence, injectivity and surjectivity results persist in analogy, with surjectivity linked to $\gamma(\widetilde{{\boldsymbol{M}}})=\infty$. The results unify and extend prior Stieltjes-moment analyses in ultradifferentiable and Gelfand-Shilov contexts, clarifying how growth properties of weight sequences govern solvability and the construction of right inverses.

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.

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

TL;DR

This work extends the Stieltjes moment problem to Gelfand-Shilov spaces defined by weight sequences, introducing a larger target space when the weaker (sm) condition holds. It develops a Laplace/Borel-type framework linking the moment mapping to asymptotic Borel results, proving injectivity criteria via and surjectivity criteria via (and local/global right inverses with uniform -scaling). When (sm) fails, it introduces a new target sequence and shows that, under fast, regular growth making a weight sequence, injectivity and surjectivity results persist in analogy, with surjectivity linked to . The results unify and extend prior Stieltjes-moment analyses in ultradifferentiable and Gelfand-Shilov contexts, clarifying how growth properties of weight sequences govern solvability and the construction of right inverses.

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.

Paper Structure

This paper contains 8 sections, 26 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ultraholomorphic classes on the upper half-plane and the asymptotic Borel mapping
  4. Gelfand-Shilov spaces and the Fourier transform
  5. The Laplace transform
  6. A modified Stieltjes moment problem in Gelfand-Shilov spaces
  7. The situation when $\operatorname{(sm)}$ fails to hold
  8. Some comments on the condition $\operatorname{(sm)}$ and the sequence $\widetilde{{\boldsymbol{M}}}$

Key Result

Lemma 2.3

Let ${\boldsymbol{M}}$ be a sequence and let $f \in \mathcal{A}_{{\boldsymbol{M}},h}(\mathbb{H})$ for some $h > 0$. Then, exists for all $x \in \mathbb{R}$ and $p \in \mathbb{N}_0$. Moreover, $f_0 \in C^\infty(\mathbb{R})$, $f_0^{(p)} = f_p$ for all $p \in \mathbb{N}$, and

Theorems & Definitions (56)

  • Remark 2.1
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • Theorem 2.6: momentsdebrouwere, Theorem 7.4.(b)
  • Theorem 2.7
  • Proposition 2.8
  • proof
  • Remark 2.9
  • Proposition 2.10
  • ...and 46 more