For which $f \in S(R^n)$, the Schwartz class, $\hat f \in D(R^n)$ ?
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GelfandShilov, Generalized Functions, Vol. 1 gives a rather extended discussion of this function class, as well as the corresponding class of distributions. 


Hmmm, Paley wiener provides a way through it. Since we have the result, $\check f $ = $\hat \hat f$ (This result holds for $S(R^n)$,as periodicity of Fourier transform is 4) Let $C$ ={ $ f \in S(R^n)$  $\hat f \in D(R^n)$ } Then we know that $f \in C$, $\check f $ is an entire function with exponential bound. Then $f$ will also be entire function with exponential bound. Converse of PaleyWiener Theorem will ensure that this is exactly the class. I hope I didn't make any mistake. :D 

