Non-smooth continuous and compact-supported 1-D functions with Fourier transform: Could they be defined through differential equations? Any examples?
Motivation
I have learned recently here that non analytic function could be compact-supported (except from the zero function), and also that no non-smooth function could be analytical, and also here that no finite-duration function could be linear, so finite-duration functions cannot be the solution of an ordinary linear differential equation, neither be solved through the framework of LTI systems (Linear-Time-Invariant), so thinking that any continuous-time "naive" physical phenomena should be time-limited, this tells that everything I have seen in engineering are only approximations.
So thinking in "naive" physical systems described by 1-Dimension functions $f(t)$ that are:
- $f(t)$ is continuous, so no teleportation is allowed. Also the system is of continuous-time, and $f(t) \in \mathbb{R}$ (since every differentiable complex function is analytic, lets start by restricting the function to the reals).
- $f(t)$ is of finite-duration (time-limited), so is a compact-supported function (here these conditions are equivalent because is a 1D function), and since is continuous, is also a bounded function $\|f\|_\infty < \infty$.
- $f(t)$ has a Fourier Transform, so it has a frequency spectra. This condition actually have many more restrictions hidden within: for having a convergent Fourier Transform, the function $f(t)$ must be absolutely integrable $\|f\|_1 < \infty$, and since is bounded, it is also of finite energy $\|f\|_2 < \infty$.
- The function $f(t)$ is "well-behaved", so nowhere-differentiable functions as the Weierstrass function or fractals are discarded (I don´t know how formally state this restriction, but I believe is related to (3) – hope you can comment).
Looking for a framework to analyze this finite-duration functions, I found here that there exist at least one compact-supported function $\varphi (t)$ with support $\mathrm{supp}(\varphi) = [-1,\,1]$, $\varphi(t) \leq 0$ and $\varphi(0)=1$ such that for any $t \in \mathbb{R}$:
$$\varphi'(t) = 2\varphi(2t+1)-2\varphi(2t-1)$$
where is important to note again that the differential equation is defined for any time and not only the support. This function $\varphi(t)$ is also a "bump function" $\in C_c^\infty$, so is also a smooth function, which need to fulfill at the boundaries of its support $\partial t = \{-1,\,1\}$ the function $\varphi(-1) = \varphi(1) = 0$ and $\lim\limits_{t \to \partial t^{\pm}} \frac{d^n\varphi(t)}{dt^n} = 0,\,\forall n\geq 0,\,n \in \mathbb{Z}$ to keep smoothness.
But thinking in modeling a ball that is traveling in straight line and crashed with a wall (elastic collision), it will experience a sudden change in velocity, having a "sharp-edge" in the function that describes its position $f(t)$ (like the "absolute-value" function $f(x) = |x|$), which will traduce in a bounded-jump-discontinuity in $\|f'(t)\|_\infty < \infty$, that will become in an unbounded-discontinuity on acceleration $\|f''(t)\|_\infty \to \infty$, so at least for a "naive" physical system:
- $f(t)$ cannot be an smooth function, so $f(t) \notin C_c^\infty$, and so, $f(t)$ cannot be an analytical function int the whole domain $t\in\mathbb{R}$ – as it was every solution of physical systems I saw in engineering, that can be solved at least through power series like Taylor expansions. Note that it could be analytical "within" the compact-support without including its edges.
So I am looking for a mathematical framework to work and analyze this kind of functions and the dynamical systems they can follow (references with closed-form examples if it's possible):
A) Functions that fulfill (1) to (5): Could they be defined through differential equations? Any examples?
Here meaning that the differential equation (probably non-linear), is defined for any $t \in \mathbb{R}$ even where its solution has a compact support.
I already know that if the domain is split piecewise, every ordinary linear differential equation within a selected compact-supported-piece of the domain could be solved through the finite-duration Fourier Transform as I done here, where even the effect of the discontinuities at the edges of the compact-support $\partial t$ on the Fourier Transform could be avoided, so Parseval's relation keep being hold. Also, with this is not needed to have a solution that starts and ends at zero at the edges $\partial t$, but if you feel more comfortable adding these requirement to the framework, it is not a problem (since can easily been fixed later with the transform $\mathring{\mathbb{F}}\{\cdot\}$).
But because of this, is clear that this splitting of the domain don´t solve the differential equation, it just take a compact-supported piece of an already known solution, that is why I believe is needed to the equation to be defined for the whole $\forall t \in \mathbb{R}$ even when their solution is compact-supported.
As example, the equation of the standard damped pendulum with friction $\ddot{\theta}(t)+a\dot{\theta}(t)+b\sin(\theta(t)) = 0$ which has not known exact closed-form solution: Could be so because $\theta(t)$ has to be compact-supported?
Hope you understand what I am asking for, and please, give examples of functions/equations in your answers (I am not formally trained in these "kind of math things", and I have been learning through questions here on SE, so my knowledge is highly limited).
So far, I have not found How to work with these finite-duration functions/systems, even when they are, I think, the easier-to-find kind of physical phenomena, and given this, surprisingly (at least for me), I haven´t found any "widely-known theoretical framework" specifically related to these functions.
I am specially interested in founding if being of finite-duration will rise some restrictions to the maximum rate of change of the function $f(t)$, like "since $f(t)$ is of finite-duration, then, $\|f'\|_\infty < \infty$", but I cannot found a framework to analyze causality of finite-duration functions yet (I only known the conditions for LIT systems and they not apply here), like functions like $f(t) = \sin(|t|\pi),\,|t|\leq 1$ are allowed, but functions like $f(t) = t\log(|t|),\,|t|\leq 1$ are not because they will violate every possible finite-causal-speed model since $\sup_t |f'(t)| \to \infty$ (this specific question is asked here). So any related reference will be also appreciated.
Beforehand, thanks you very much.
Added later
I have found later this paper that analyze finite-time controllers as continuous time differential equations with finite duration, but it looks like that are solution that have only an "end", and I am not sure if they have also a beginning (so, being properly compact-supported solutions). Also, is too advanced for me to fully understand what is being said (I have no clue about Lyapunov theory). Hope you can review it to tell me if the solutions of the examples are compact-supported (so they become examples of what I am looking for), or not, and maybe someone can explain it in simple.
Best Answer
On this other question it was posted an answer where is shown the Norton's Dome example, which is not a finite duration solution in its standard form, but its differential equation indeed stands finite duration solutions: $$\ddot{r} = \sqrt{r}$$ stands the final duration solutions: $$r(t) = \frac{1}{144}\left(T-t\right)^4\theta(T-t)$$ with $\theta(t)$ the standard unitary step function (Heaviside's function), and an ending time $T$ dependent of the initial conditions: $T=\sqrt[4]{144\,r(0)}>0$ for $r(0)>0$.
And with Wolfram-Alpha it is possible to evaluate is finite duration Fourier transform giving: $$\int\limits_{0}^{T} \frac{1}{144}\left(T-t\right)^4\theta(T-t)e^{-iwt}\,dt = \frac{-iT^2w^2+2Tw-2ie^{-iTw}+2i}{144w^3}$$