MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For which $f \in S(R^n)$, the Schwartz class, $\hat f \in D(R^n)$ ?

share|cite|improve this question
Note that such functions must be real analytic, and take a look at the Paley-Wiener theorem. – Willie Wong Mar 9 '11 at 14:05
For instance, check Theorem 7.3.1 in the first volume of Lars Hormander's treatise – Piero D'Ancona Mar 9 '11 at 17:19
Just to expand Willie Wong's comment slightly: the Paley-Wiener Theorem exactly characterises the inverse Fourier transform of $L^2(-A,A)$. Since the Schwartz function $\widehat{f}$ is compactly supported if and only if it is in $L^2(-A,A)$ for some $A$, you immediately get the answer: $f$ extends to an entire function of exponential growth, which is $L^2$ on every horizontal line. See e.g. Wikipedia: I quite like the treatment in Koosis, Theory of $H_p$ spaces; many books on Hardy Spaces give it. – Zen Harper Mar 13 '11 at 12:02
Let me insist with Hormander's version :)) If you read through the book, you will see that the true spirit of the result is not exactly compact support vs. exponential growth, but, more precisely: if the support is contained in a closed convex set, then you can bound the growth of the F.transform in the dual directions, and viceversa – Piero D'Ancona Mar 29 '11 at 8:48

Gelfand-Shilov, Generalized Functions, Vol. 1 gives a rather extended discussion of this function class, as well as the corresponding class of distributions.

share|cite|improve this answer

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 Paley-Wiener Theorem will ensure that this is exactly the class. I hope I didn't make any mistake. :D

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.