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

Question

Are there any time-limited and continuous one-variable functions $f(t)$ with bounded derivative $||f'(t)||_\infty < \infty$ (not meaning here they are also necessarily differentiable), such as their Fourier transform $F(w)$ makes diverge the following integral $\int\limits_{-\infty}^\infty|iwF(w)+f(t_F)\,e^{-iwt_F}-f(t_0)\,e^{-iwt_0}|\,dw \to \infty$?? Or these kind of functions are an empty set (for each of the following scenarios)?

The different terms from the questions of the tittle are just for avoiding the effects of the discontinuity on the edges of the compact-support $\partial t = \{t_0,\,t_F\}$ (starting and ending times), since they introduce Dirac's Delta functions $\delta(t)$ in the derivative $f'(t)$ ("artificially" in my opinion, since to model time limited phenomena I am interested only in what is happening "within" the compact support).

If you feel uncomfortable with them, just assume also that the functions $f(t)$ begins and finishes at zero $f(t_0)=f(t_F)=0$. From the following, I will use both definitions as equivalent since the problem is avoidable (I explained one way to overcome it here). Please keep it in mind, or it will make harder to find counterexamples since this edges-discontinuities will make the standard $\int_{-\infty}^\infty|wF(w)|dw$ always diverge, since the derivative will be unbounded because of these delta functions, as I will explain now.

I am trying to understand the figure of the integral $\int_{-\infty}^\infty|wF(w)|dw$ which is an upper bound for the maximum rate of change of the function $f(t)$: $$ \sup\limits_t \left| f'(t)\right| \leq \int_{-\infty}^\infty|wF(w)|dw$$ It has an individual name? (as the Dirichlet Energy, as example), this for being able to look for its properties by myself. Any references are welcome.

Directly from the inequality I know that if the derivative is unbounded $||f'(t)||_\infty \to \infty \Rightarrow \int_{-\infty}^\infty|wF(w)|dw \to \infty$ will always diverge, and conversely, if "this" integral is bounded $\int_{-\infty}^\infty|wF(w)|dw < \infty \Rightarrow ||f'(t)||_\infty < \infty$ the maximum rate of change will be bounded (even when time-limited functions has unlimited bandwidth on the frequencies), but I want to know if there exists any cases of functions that lie in-between these two scenarios (I have already looked unsuccessfully for counterexamples by myself).

I am specially interested in these five scenarios (from less to more restrictive - I believe):

1. General time-limited and continuous one-variable functions $f(t)$, as is already asked
2. Time-limited continuous one-variable functions which are also absolutely integrable $\int\limits_{t_0}^{t_F}|f(t)|\,dt < \infty$ and energy finite $\int\limits_{t_0}^{t_F}|f(t)|^2 dt < \infty$
3. Functions that fulfill (1) and (2) and are also have their absolute value of its Fourier Transform bounded $\int\limits_{-\infty}^{\infty} |F(w)| dw < \infty$
4. Functions that fulfill (1) to (3) and also have finite Dirichlet Energy $\int\limits_{t_0}^{t_F} |f'(t)|^2 dt < \infty$
5. Functions that fulfill (1) to (4) and there also of bounded total variation $V_{[t_0,\,t_F]}(f(t)) < \infty$

I want to know if any of these intermediate conditions stages makes the integral $\int_{-\infty}^\infty|wF(w)|dw$ becomes bounded, or if are totally unrelated.

Please notice that neither of these conditions are requiring to $f(t)$ to be differentiable. But I am not interested in "bad-behaved" things like nowhere-differentiable functions as Brownian motions, or fractals, or Cantor or Weierstrass functions, and things like that (at least not this time)

Any counterexample will be welcome either. Beforehand, thanks you very much.

Answer 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.

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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.