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@ARTICLE{Jakobsen:491605,
      author       = {Jakobsen, Mads Bregenholt and Feichtinger, Hans G.},
      title        = {{T}he inner kernel theorem for a certain {S}egal algebra},
      reportid     = {PUBDB-2023-00297, arXiv:1806.06307},
      year         = {2022},
      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.},
      cin          = {FS-SC / U HH},
      cid          = {I:(DE-H253)FS-SC-20210408 / $I:(DE-H253)U_HH-20120814$},
      pnm          = {623 - Data Management and Analysis (POF4-623)},
      pid          = {G:(DE-HGF)POF4-623},
      experiment   = {EXP:(DE-MLZ)NOSPEC-20140101},
      typ          = {PUB:(DE-HGF)25},
      eprint       = {1806.06307},
      howpublished = {arXiv:1806.06307},
      archivePrefix = {arXiv},
      SLACcitation = {$\%\%CITATION$ = $arXiv:1806.06307;\%\%$},
      url          = {https://bib-pubdb1.desy.de/record/491605},
}