Representation Theory of Reductive Groups

Volume 40 of the series Progress in Mathematics pp 1-19

Multipliers and a Paley-Wiener Theorem for Real Reductive Groups

  • James Arthur

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The classical Paley-Wiener theorem is a description of the image of Cc (IR) under Fourier transform. The Fourier transform
$$\rm{\hat f (\wedge)\ =\ \int^\infty_{-\infty}\ f(x)e^{\wedge x}\ dx}$$
is defined a priori for purely imaginary numbers ⋀, but if f has compact support \(\hat f\) will extend to an entire function on the complex plane.