Journal Article

PUBDB-2022-01667

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png The inner kernel theorem for a certain Segal algebra

Feichtinger, H. G. ; Jakobsen, M. B. (Corresponding author)DESY*

2022
Springer Wien [u.a.]

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Please use a persistent id in citations: doi:10.1007/s00605-022-01702-4  doi:10.3204/PUBDB-2022-01667

Report No.: arXiv:1806.06307

Abstract: The Segal algebra $\mathbf{S}_{0}(G)$ is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups $G$. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from $\mathbf{S}_{0}(G_{1})$ to $\mathbf{S}'_{0}(G_{2})$ by generalized functions in $\mathbf{S}'_{0}(G_{1} \times G_{2})$ the 'outer kernel theorem'. The main subject of this paper is to formulate what we call the 'inner kernel theorem'. This is the characterization of those linear operators that have kernels in $\mathbf{S}_{0}(G_{1} \times G_{2})$. Such operators are regularizing -- in the sense that they map $\mathbf{S}'_{0}(G_{1})$ into $\mathbf{S}_{0}(G_{2})$ in a $w^{*}$ to norm continuous manner. A detailed functional analytic treatment of these operators is given and applied to the case of general LCA groups. This is done without the use of Wilson bases, which have previously been employed for the case of elementary LCA groups. We apply our approach to describe natural laws of composition for operators that imitate those of linear mappings via matrix multiplications. Furthermore, we detail how these operators approximate general operators (in a weak form). As a concrete example, we derive the widespread statement of engineers and physicists that pure frequencies 'integrate' to a Dirac delta distribution in a mathematically justifiable way.

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Contributing Institute(s):

1. Scientific computing (FS-SC)
2. Uni Hamburg (U HH)

Research Program(s):

1. 623 - Data Management and Analysis (POF4-623) (POF4-623)

Experiment(s):

1. No specific instrument

Appears in the scientific report 2022

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; ; ; Clarivate Analytics Master Journal List ; Current Contents - Physical, Chemical and Earth Sciences ; Essential Science Indicators ; IF < 5 ; JCR ; SCOPUS ; Science Citation Index Expanded ; Web of Science Core Collection

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Linked articles:

http://join2-wiki.gsi.de/foswiki/pub/Main/Artwork/join2_logo100x88.png Preprint Jakobsen, M. B. (Corresponding author)DESY* ; Feichtinger, H. G.
The inner kernel theorem for a certain Segal algebra
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