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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)

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
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