000491605 001__ 491605
000491605 005__ 20230214113043.0
000491605 0247_ $$2arXiv$$aarXiv:1806.06307
000491605 0247_ $$2altmetric$$aaltmetric:43856183
000491605 037__ $$aPUBDB-2023-00297
000491605 041__ $$aEnglish
000491605 088__ $$2arXiv$$aarXiv:1806.06307
000491605 1001_ $$0P:(DE-H253)PIP1098744$$aJakobsen, Mads Bregenholt$$b0$$eCorresponding author$$udesy
000491605 245__ $$aThe inner kernel theorem for a certain Segal algebra
000491605 260__ $$c2022
000491605 3367_ $$0PUB:(DE-HGF)25$$2PUB:(DE-HGF)$$aPreprint$$bpreprint$$mpreprint$$s1673964033_10756
000491605 3367_ $$2ORCID$$aWORKING_PAPER
000491605 3367_ $$028$$2EndNote$$aElectronic Article
000491605 3367_ $$2DRIVER$$apreprint
000491605 3367_ $$2BibTeX$$aARTICLE
000491605 3367_ $$2DataCite$$aOutput Types/Working Paper
000491605 520__ $$aThe Segal algebra $\mathbf{S}_{0}(G)$ is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups $G$. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from $\mathbf{S}_{0}(G_{1})$ to $\mathbf{S}'_{0}(G_{2})$ by generalized functions in $\mathbf{S}'_{0}(G_{1} \times G_{2})$ the 'outer kernel theorem'. The main subject of this paper is to formulate what we call the 'inner kernel theorem'. This is the characterization of those linear operators that have kernels in $\mathbf{S}_{0}(G_{1} \times G_{2})$. Such operators are regularizing -- in the sense that they map $\mathbf{S}'_{0}(G_{1})$ into $\mathbf{S}_{0}(G_{2})$ in a $w^{*}$ to norm continuous manner. A detailed functional analytic treatment of these operators is given and applied to the case of general LCA groups. This is done without the use of Wilson bases, which have previously been employed for the case of elementary LCA groups. We apply our approach to describe natural laws of composition for operators that imitate those of linear mappings via matrix multiplications. Furthermore, we detail how these operators approximate general operators (in a weak form). As a concrete example, we derive the widespread statement of engineers and physicists that pure frequencies 'integrate' to a Dirac delta distribution in a mathematically justifiable way.
000491605 536__ $$0G:(DE-HGF)POF4-623$$a623 - Data Management and Analysis (POF4-623)$$cPOF4-623$$fPOF IV$$x0
000491605 588__ $$aDataset connected to arXivarXiv
000491605 693__ $$0EXP:(DE-MLZ)NOSPEC-20140101$$5EXP:(DE-MLZ)NOSPEC-20140101$$eNo specific instrument$$x0
000491605 7001_ $$0P:(DE-HGF)0$$aFeichtinger, Hans G.$$b1
000491605 8564_ $$uhttps://bib-pubdb1.desy.de/record/491605/files/1806.06307v3.pdf$$yRestricted
000491605 8564_ $$uhttps://bib-pubdb1.desy.de/record/491605/files/1806.06307v3.pdf?subformat=pdfa$$xpdfa$$yRestricted
000491605 909CO $$ooai:bib-pubdb1.desy.de:491605$$pVDB
000491605 9101_ $$0I:(DE-588b)2008985-5$$6P:(DE-H253)PIP1098744$$aDeutsches Elektronen-Synchrotron$$b0$$kDESY
000491605 9131_ $$0G:(DE-HGF)POF4-623$$1G:(DE-HGF)POF4-620$$2G:(DE-HGF)POF4-600$$3G:(DE-HGF)POF4$$4G:(DE-HGF)POF$$aDE-HGF$$bForschungsbereich Materie$$lMatter and Technologies$$vData Management and Analysis$$x0
000491605 9141_ $$y2022
000491605 915__ $$0StatID:(DE-HGF)0580$$2StatID$$aPublished
000491605 9201_ $$0I:(DE-H253)FS-SC-20210408$$kFS-SC$$lScientific computing$$x0
000491605 9201_ $$0I:(DE-H253)U_HH-20120814$$kU HH$$lUni Hamburg$$x1
000491605 980__ $$apreprint
000491605 980__ $$aVDB
000491605 980__ $$aI:(DE-H253)FS-SC-20210408
000491605 980__ $$aI:(DE-H253)U_HH-20120814
000491605 980__ $$aUNRESTRICTED