The vector-valued Stieltjes moment problem with general exponents

We characterize the sequences of complex numbers ( z n ) n ∈ N and the locally complete ( DF )-spaces E such that for each ( e n ) n ∈ N ∈ E N there exists an E -valued function f on ( 0 , ∞ ) (satisfying a mild regularity condition) such that ∫ 0 ∞ t z n f ( t ) d t = e n , ∀ n ∈ N , where the inte...

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Bibliographic Details
Published inBanach journal of mathematical analysis Vol. 18; no. 3
Main Authors Debrouwere, Andreas, Neyt, Lenny
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.07.2024
Subjects
Functional Analysis
Mathematics
Mathematics and Statistics
Operator Theory
Original Paper
Vector-valued integration
46G10
46E10
Weighted spaces of (vector-valued) smooth functions
47A57
30E05
46E40
46A63
Stieltjes moment problem
Linear topological invariants
44A60
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Summary:We characterize the sequences of complex numbers ( z n ) n ∈ N and the locally complete ( DF )-spaces E such that for each ( e n ) n ∈ N ∈ E N there exists an E -valued function f on ( 0 , ∞ ) (satisfying a mild regularity condition) such that ∫ 0 ∞ t z n f ( t ) d t = e n , ∀ n ∈ N , where the integral should be understood as a Pettis integral. Moreover, in this case, we show that there always exists a solution f that is smooth on ( 0 , ∞ ) and satisfies certain optimal growth bounds near 0 and ∞ . The scalar-valued case ( E = C ) was treated by Durán (Math Nachr 158:175–194, 1992). Our work is based upon his result.
ISSN:2662-2033
1735-8787
DOI:10.1007/s43037-024-00364-8