Harmonic Analysis for Function Fields

Question

Hi,

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: http://www.springerlink.com/content/g45268h62705h470/

It should contain some of these: the additive and multiplicative Fourier transform, the Haarmeasures, the Gamma functions, etc.

Answer 1

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

I. M. Gelfand, M. I. Graev, and I. I. Piatetski-Shapiro, Representation Theory and Automorphic Functions, Saunders, Philadelphia, 1969.

For some more recent results see also

A. N. Kochubei, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields, Marcel Dekker, New York, 2001.

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

A. N. Kochubei, Analysis in Positive Characteristic, Cambridge University Press, 2009.