Abstract
We study the nuclearity of the Gelfand–Shilov spaces \({\mathcal {S}}^{({\mathfrak {M}})}_{({\mathscr {W}})}\) and \({\mathcal {S}}^{\{{\mathfrak {M}}\}}_{\{{\mathscr {W}}\}}\), defined via a weight (multi-)sequence system \({\mathfrak {M}}\) and a weight function system \({\mathscr {W}}\). We obtain characterizations of nuclearity for these function spaces that are counterparts of those for Köthe sequence spaces. As an application, we prove new kernel theorems. Our general framework allows for a unified treatment of the Gelfand–Shilov spaces \({\mathcal {S}}^{(M)}_{(A)}\) and \({\mathcal {S}}^{\{M\}}_{\{A\}}\) (defined via weight sequences M and A) and the Beurling–Björck spaces \({\mathcal {S}}^{(\omega )}_{(\eta )}\) and \({\mathcal {S}}^{\{\omega \}}_{\{\eta \}}\) (defined via weight functions \(\omega\) and \(\eta\)). Our results cover anisotropic cases as well.
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Funding
A. Debrouwere was supported by FWO-Vlaanderen through the postdoctoral Grant 12T0519N. L. Neyt gratefully acknowledges support by Ghent University through the BOF-Grant 01J11615. J. Vindas was supported by Ghent University through the BOF-Grants 01J11615 and 01J04017.
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Debrouwere, A., Neyt, L. & Vindas, J. The nuclearity of Gelfand–Shilov spaces and kernel theorems. Collect. Math. 72, 203–227 (2021). https://doi.org/10.1007/s13348-020-00286-2
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DOI: https://doi.org/10.1007/s13348-020-00286-2
Keywords
- Gelfand–Shilov spaces
- Nuclear spaces
- Schwartz kernel theorems
- Weight sequence systems
- Weight function systems