Schwartz’ kernel theorem for Fourier hyperfunctions. (English) Zbl 1044.46531
Begehr, Heinrich G. W. (ed.) et al., Proceedings of the second ISAAC congress. Vol. 1. Proceedings of the International Society for Analysis, its Applications and Computation congress, Fukuoka, Japan, August 16–21, 1999. Dordrecht: Kluwer Academic Publishers (ISBN 0-7923-6597-6/hbk). Int. Soc. Anal. Appl. Comput. 7, 743-754 (2000).
L. Schwartz’ kernel theorem, although a rather abstract statement about continuous bilinear forms on test function spaces, has found numerous applications. In particular, it enables the presentation of \(n\)-point correlation functions in axiomatic quantum field theory by a concrete generalised function, namely the kernel. E. Brüning together with S. Nagamachi have used this theorem for the case of Fourier hyperfunctions many times in their account of hyperfunction quantum field theory [J. Math. Phys. 42, No. 1, 99–129 (2001; Zbl 1013.81034), Lett. Math. Phys. 63, No.2, 141-155 (2003; Zbl 1029.81048)]. The present paper is based on a talk at the 1999 ISAAC conference and gives a gentle, pedagogical account of a concise, functional-analytic proof of the kernel theorem for Fourier hyperfunctions. A full treatment can be found in the corresponding article of E. Brüning and S. Nagamachi [J. Math. Univ. Tokushima 35, 57–64 (2001; Zbl 1056.46038)].
For the entire collection see [Zbl 1022.00009].
For the entire collection see [Zbl 1022.00009].
Reviewer: Andreas Schmidt (Frankfurt am Main)
MSC:
| 46F15 | Hyperfunctions, analytic functionals |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
