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Abstract

Journal Information: ID: Results Math Title: Results in Mathematics ISSN: 1422-6383, 1420-9012 Publisher: Springer International Publishing Location: Cham Title: The Stieltjes Moment Problem in Gelfand-Shilov Spaces Defined by Weight Sequences in the Absence of Derivation Closedness Authors: - Javier Jiménez-Garrido - Ignacio Miguel-Cantero - Javier Sanz - Gerhard Schindl 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

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The paper characterizes when the Stieltjes moment mapping is injective or surjective in Gelfand‑Shilov spaces defined by weight sequences, showing it is never bijective under common conditions.

What’s new

i.Introduce a larger target space for the Stieltjes moment mapping within Gelfand‑Shilov spaces defined by weight sequences
ii.Provide injectivity and surjectivity criteria for the moment mapping using weight‑sequence conditions
iii.Show that the moment mapping is never bijective when the weight sequence satisfies the derivation‑closed (dc) condition
iv.Extend results to general weight sequences when the shifted‑moments condition fails

How they did it

The authors work in Gelfand‑Shilov spaces defined via weight sequences, analyse the moment mapping through properties such as logarithmic convexity, derivation‑closedness and shifted moments, and employ tools like the Laplace transform, optimal flat functions and asymptotic Borel mapping to derive injectivity and surjectivity conditions.

What they found

Injectivity holds iff ∑_{p=0}^∞ 1/m_p^{1/2}=∞ (Theorem 2.7)
Surjectivity holds iff γ_{M}>2 (Theorem 2.6)
The moment mapping is never bijective when the weight sequence satisfies the dc condition (Corollary 4.7)

Caveats

✕Results depend on the weight sequence satisfying specific conditions (e.g., dc, sm, shifted‑moments); cases outside these conditions are only partially addressed
✕The non‑bijectivity conclusion requires the dc condition; the behavior for sequences lacking dc remains open
✕The framework does not treat weight sequences that fail the sm condition, limiting applicability

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