Is there some connection between the power-spectrum of a real function $f:\mathbb{R}\to\mathbb{R}$ (that is, its Fourier transform) and the convergence radius of its Taylor expansion around arbitraty $x\_0$? Intuitively, I would expect a function with 'limited power at high frequencies' to have 'large convergence radius' around each point, but I could not find such result.


Thanks!



Taylor expansion convergence relation to power-spectrum
A basic phenomenon in the direction of your question is the following. If $\hat{f}(\xi)e^{C |\xi|}$ ($C<+\infty$) is integrable, then $f$ has a holomorphic extension to the strip of width $C$ around the $x$-axis. As a consequence, the Taylor series of $f$ converges on an interval of radius $C$ around each point. One can generalize this to several variables.


