Definition of Fourier transform

Question

Is there a difference between $\hat{f}(y)=\int_{\mathbb{R}}f(x)e^{-2\pi ixy}dx$ and $\hat{f}(y)=\int_{\mathbb{R}}f(x)e^{2\pi ixy}dx$ or is it only convention?

Answer 1

Yes and yes. Yes, there is a difference: The obvious sign. But no, the difference is not so important as to have a significant impact on the important theorems about Fourier transforms (it adds a sign to some equations). Therefore it is mostly convention which definition you use and most of the time is is easy to derive how to go from one equation to the other. The former seems to be more common as far as I can tell.