Abstract
In order to understand all completions of the field ℚ of rational numbers at the same time, one forms a product of all completions, which needs to me modified in order to form a locally compact space. These modified products are discussed at length and in very general terms until they are applied to the situation at hand. This yields the ring of adeles. Its unit group is called the group of ideles. Fourier analysis on both of them is studied in quite explicit terms.
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References
Deitmar, A.: A First Course in Harmonic Analysis, 2nd edn. Universitext. Springer, New York (2005)
Deitmar, A., Echterhoff, S.: Principles of Harmonic Analysis. Universitext. Springer, New York (2009)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, vol. 32. Princeton University Press, Princeton (1971)
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Deitmar, A. (2013). Adeles and Ideles. In: Automorphic Forms. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4435-9_5
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DOI: https://doi.org/10.1007/978-1-4471-4435-9_5
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4434-2
Online ISBN: 978-1-4471-4435-9
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