# Explicit Paley-Wiener function

By a Paley-Wiener function I mean a function $f(z)$ that is the Fourier image of a test function. Equivalently, by Paley-Wiener theorem, $f(z)$ is an entire function that is of rapid decay on the real line and has a finite exponential type, $$|f(z)|\le\frac{C_ne^{B|\mathrm{Im}z|}}{1+|\mathrm{Re}z|^n},\quad\forall n\in\mathbb{N}.$$

Question: What is the simplest explicit formula for a non-zero Paley-Wiener function?

The best I can think of right now is $$f(z)=\int_0^1dte^{-\imath zt-\frac1{t(1-t)}}$$ which is not really explicit. Thanks.

• The awkwardness that seems necessary to express a test function seems inescapable, indeed, ... Mar 25, 2018 at 0:47

## 1 Answer

In the standard terminology, a Paley-Wiener function is a Fourier transform of a function from $L^2$ with compact support.

The simplest formula for a Paley-Wiener function is $\phi(x)=(\sin x)/x$. This is the Fourier transform of the characteristic function of an interval.

But you seem to have some unusual definition of a Paley Wiener function, though you do not state precisely what your "test functions" are. If you mean infinitely differentiable functions with compact support, take an infinite product of the form $\prod \phi(a_kx)$ where $a_k$ is an appropriately chosen sequence of real numbers tending to $0$. I don't think there is a simpler form, without an infinite product or an integral.

• Thanks for the answer. I am not aware of any context where a characteristic function is classified as a test function. I think the term test function is more or less commonly agreed upon, see the Wiki page. The term Paley-Wiener space is more ambivalent, therefore I gave a precise definition. It is pretty clear that most probably there is no simple such function, but one wants to know why, and what is the simplest among all known.Thanks for the idea of infinite products. I will try to see what it gives May 22, 2017 at 16:01
• Although I've not taken the trouble to "check around", I am mildly surprised at the usage of "Paley-Wiener function" for Fourier transform of compactly-supported $L^2$ function. Namely, for me, it means "FT of test function". (And, yes, at another extreme, there is Schwartz' result about compactly-supported distributions.) So, I ask a sincere question about how widespread this usage is... Mar 25, 2018 at 0:46