Harmonic Analysis for Function Fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:36:43Z http://mathoverflow.net/feeds/question/61518 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61518/harmonic-analysis-for-function-fields Harmonic Analysis for Function Fields Marc Palm 2011-04-13T08:07:00Z 2011-04-13T09:15:15Z <p>Hi,</p> <p>Where goes a characterictic $0$ person, in order to learn about the local harmonic analysis for local fields in characteristic $p$? Is there nice and conscise reference for the local fields in positive characteristic. Something conscise like Sally's survey, which is for characterictic zero only: <a href="http://www.springerlink.com/content/g45268h62705h470/" rel="nofollow">http://www.springerlink.com/content/g45268h62705h470/</a></p> <p>It should contain some of these: the additive and multiplicative Fourier transform, the Haarmeasures, the Gamma functions, etc.</p> http://mathoverflow.net/questions/61518/harmonic-analysis-for-function-fields/61526#61526 Answer by Anatoly Kochubei for Harmonic Analysis for Function Fields Anatoly Kochubei 2011-04-13T09:15:15Z 2011-04-13T09:15:15Z <p>Harmonic analysis dealing with complex-valued functions on a local field $K$ depends very little on the characteristic of $K$ and can be given in the general case, for an arbitrary $K$. See </p> <p>I. M. Gelfand, M. I. Graev, and I. I. Piatetski-Shapiro, Representation Theory and Automorphic Functions, Saunders, Philadelphia, 1969.</p> <p>For some more recent results see also</p> <p>A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields, Marcel Dekker, New York, 2001.</p> <p>However, if one is interested in analysis of functions $K\to K$, the theory in positive characteristic is completely different from the $p$-adic case. See</p> <p>A. N. Kochubei, Analysis in Positive Characteristic, Cambridge University Press, 2009. </p>