Geodesic
Distributions and Mnemofunctions on Adeles. Fourier Transform
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 228-240

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Some classical results are recalled, and a finite part distribution is interpreted as the zero-order term in the expansion of a homogeneous distribution. An adelic finite part distribution and a generalization of the Tate distribution are defined, and their Fourier transforms are calculated. The machinery of mnemofunctions (nonlinear generalized functions) is adapted to $p$-adic and adelic cases, and product formulas for some specific distributions are given. 
@article{TM_2004_245_a22,
     author = {E. M. Radyno and Ya. V. Radyno},
     title = {Distributions and {Mnemofunctions} on {Adeles.} {Fourier} {Transform}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {228--240},
     publisher = {mathdoc},
     volume = {245},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_245_a22/}
}
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E. M. Radyno; Ya. V. Radyno. Distributions and Mnemofunctions on Adeles. Fourier Transform. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 228-240. http://geodesic.mathdoc.fr/item/TM_2004_245_a22/