Divergent Fourier Analysis using degrees of observability

Abstract

The aim of this work is to generalize the methods of Fourier Analysis in order to apply them to a wide class of possibly non-integrable functions, with infinitely many variables. The method consists in distinguishing several levels of observability, with a natural meaning. Mathematical coherence is ensured by the fact that these natural concepts are represented within a sure mathematical framework, that of the relative set theory [Y. Péraire, Théorie relative des ensembles internes, Osaka J. Math. 29 (1992) 267–297; Y. Péraire, Some extensions of the principles of idealization transfer and choice in the relative internal set theory, Arch. Math. Logic 34 (1995) 269–277]. This work is also a step for another approach of the Fourier transform of functionals. It can be related to the one, which use double extensions of standard real numbers, performed by T. Nitta and T.Okada in [T. Okada, T. Nitta, Infinitesimal Fourier transformation for the space of functionals, in: Topics in Almost Hermitian Geometry and Related Fields, World Sci. Publ., 2005; T. Okada, T. Nitta, Poisson summation formula for the space of functionals, Nihonkai Math. J. 16 (2005) 1–21].