Talk:Paley–Wiener theorem

Maybe there is an issue on a proposition
At the end of the section "1. Holomorphic Fourier transforms" it is said: [quote]: "Conversely, any entire function of exponential type A which is square-integrable over horizontal lines is the holomorphic Fourier transform of an L2 function supported in [−A, A]."

But I believe this proposition is false, an the Gaussian function could be a counterexample: its Fourier transform it is also another Gaussian function, so both the function and its transform are Analytic functions, but neither of them are of compact support (the Gaussian function is indeed an Entire function that is square-integrable). As example the function $f(t) = e^{-t^2}$ will have a Fourier Transform $ \hat{f}(w) = \int_{-\infty}^{\infty} e^{-t^2}e^{-iwt}dt = \sqrt{\pi}e^{-\frac{w^2}{4}}$ : the only part I am not sure if this function $ \hat{f}(w) =\sqrt{\pi}e^{-\frac{w^2}{4}}$  is of exponential type or not.

Hope someone could review it and include it as example or counterexample. Beforehand thanks you very much. 200.111.172.202 (talk) 17:22, 25 July 2022 (UTC)