Are there functions $f(t)$ with $||f'(t)||_\infty < \infty$ such as their Fourier transform $F(w)$ makes $\int_{-\infty}^\infty|wF(w)|dw \to \infty$??

Solution 1:

By standard properties of the Fourier transform (check in this page the subsections 5.4 and 5.7), the condition $$\int_{-\infty}^{+\infty}| wF(w)|dw<\infty\qquad\text{(1)}$$ implies that $f’$ is continuous and bounded (and of course it has compact support due to the fact that $f$ has compact support). This means that you have counterexamples just picking a Lipschitz continuous function $f$ with discontinuous derivative, e.g., $$f(t)=e^{-|t|}-e\;,\quad t\in [-1,1].$$ What I wrote holds also for the extreme points of the interval, this means that every function that satisfies (1) can’t have non-zero derivatives in the boundary of the interval. For instance, $$f(t)=e^{-t^2}-e\;,\quad t\in [-1,1]$$ is still a counterexample, even though it is smooth inside the interval. (Off topic, this problem can be eliminated using Fourier series. First extend the function oddly around one the two extremes of the interval, then consider the fourier coefficients on this doubled interval… In this case, you can at least hope that the Fourier coefficients with respect to the doubled interval are such that $\sum |j||\hat f_j|<\infty$).

An interesting question would be: what if I assume that $f$ is continuous, compactly supported on $[t_0,t_F]$ and the derivative is also continuous at every point (including the extremes of the interval, in which the derivative has to be $0$)? Here counterexamples are harder to find. I think there still exist counterexamples, but at the moment I can’t exhibit one. I’ll try to find something here on SE.

Edit: The first answer of this post brings a nice example of a continuous compactly supported function whose fourier transform has infinite integral. I think this example could be slightly modified to give a counterexample to your question with a function with continuous derivative in every point, including the extremes.