Osipov, D. V.; Parshin, A. N. Harmonic analysis on local fields and adelic spaces. I. (English. Russian original) Zbl 1222.11137 Izv. Math. 72, No. 5, 915-976 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 5, 77-140 (2008). The authors develop the harmonic analysis in some category \(\mathcal C_{2}\) of filtered vector spaces, which contains as objects the two-dimensional local fields and the adelic spaces of algebraic surfaces over a finite field. The results include the study of certain spaces of functions and distributions on \(\mathcal C_{2}\)-objects, analogous to those introduced by Bruhat in classical harmonic analysis, the study of direct and inverse images of \(\mathcal C_{2}\)-objects, the construction of the Fourier transform on \(\mathcal C_{2}\)-objects, and the establishment of two variants of Poisson formula. Reviewer: V. Popa (Chisinau) Cited in 2 ReviewsCited in 7 Documents MSC: 11R56 Adèle rings and groups 11S31 Class field theory; \(p\)-adic formal groups 43A32 Other transforms and operators of Fourier type 43A95 Categorical methods for abstract harmonic analysis PDFBibTeX XMLCite \textit{D. V. Osipov} and \textit{A. N. Parshin}, Izv. Math. 72, No. 5, 915--976 (2008; Zbl 1222.11137); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 5, 77--140 (2008) Full Text: DOI arXiv