Which conditions must fulfill a time-limited function (so of unlimited bandwidth) $f(t)$ to have a bounded maximum rate of change?
I want to know about which conditions must fulfill a real-valued time-limited function (so compacted-supported and with unlimited bandwidth), at least in the case of one-variable $f(t)$, to have a bounded maximum rate of change $\max_t \left|\frac{d}{dt}f(t)\right|< \infty$.
I have already known that band-limited functions have bounded maximum rate of change has is explained here, and also that functions with unlimited bandwidth could achieve and infinite maximum rate of change as I explained on an answer of this question, but also, in the body of the question I also show that there exists time-limited functions as: $$ f(t) = \begin{cases} \cos^2 \left( \frac{t \pi}{2} \right) , \, \, |t| \leq 1 \\ 0, \, \, |t|>1 \end{cases}$$ which haves bounded maximum rate of change, so I want to know which conditions must fulfil an unlimited-bandwidth function to have a finite maximum slew rate (let them be named as $\text{mysterious conditions}\,\mathbb{X}$), and hopefully found a “good” approximation or upper bound for this maximum rate of change $\max_t \left|\frac{d}{dt}f(t)\right|$.
Or conversely, given my limited mathematical knowledge and since I prove myself that time-limited functions with bounded maximum slew rate do exists (it wasn´t obvious for me), at least for me, to assume that "because infinite bandwidth signals could achieve infinite max slew rate" $\Rightarrow$ "there is no conditions where under them, time-limited functions could be achieving bounded max slew rates" (so examples are just “happy coincidences”), will be falling into a logical fallacy (I believe is named Hasty generalization). So, if you could prove that these $\text{mysterious conditions}\,\mathbb{X}$ are nonexistence, it will be also great, since I could be moving out from trying to solve this problem.
Hope you can answer mathematical related answers there and physical related comments here.
Beforehand, thanks you very much.
Note: This question was migrated by somebody else from the Physics forum.
Motivation
The following part is not required to answer the question, but I would like to explain why I believe this question is an interesting one.
Since I discovered on internet that compact-supported functions (as are time-limited signals), can’t been analytic (the only analytic and compact-supported function is the zero function, as is explained here and here), I realize that every solution I have seen in engineering are approximations: every solution through standard analytic functions, every ordinary linear differential equation, every non-linear differential equation which solutions are through power series, as the Bessel, Jacobi, Hermite, Laguerre, Hypergeometric differential equations, and any function which duration is unbounded, is an approximation.
Several and interesting doubts born related to this: Could be happening some spurious effects because of these approximations? How to check for them? Is every approximation through first/second order Taylor’s series valid? Same with solutions through Perturbation Theory?... Since the electromagnetic wave equation is a linear partial differential equation which its solutions are complex exponential functions: Are Maxwell’s laws also approximations? What kind of differential equations are followed by compacted-supported functions?... Since the simplest realistic model is the damped pendulum equation with friction, which still haven´t a solution in closed form, Is this because a compacted-supported solution is required? Like changing the exponential decay of the small angle approximation by things like $ e^{x/(x-1)},\,0 \leq x \leq 1$? Or even Bump functions $\in C_c\infty$?... Or going deeply, as beautifully explained by Professor Barton Zwiebach here (min 7:25), when solving the Time-Independent Schrödinger equation for the harmonic oscillator using a Power Series ansatz, for producing solutions that doesn’t diverges to infinity on the space, i.e., to produce normalizable solutions, some restrictions have to be set on its coefficients, and these restriction are the reason why energy gets quantized – all this by solving through a power series which are analytic functions, as no finite time or finite space (local) function can be!! (this because the simplest forms of the Schrödinger equation are linear differential equations –There are also other versions, I have worked with non-linear version as the used for light propagation on fiber optics, in my case solved through numerical split-step Fourier method).
But the apparently simple question of the title has also applicable importance beyond the “philosophy” of the physics of the successful models mentioned above. Having a restricted rate of change, as in the case of bandlimited functions, could be useful to understand phenomena were the $\text{mysterious conditions}\,\mathbb{X}$ are fulfill, or even, if there any physical phenomena that could be described under these conditions, then a physical law have been also being found, maybe not a general one, but at least is the kind of things that puts your last name on Wikipedia.
As example, laser beams can be focused only up just specific “spot size”, from where they start to diverge again, phenomena that can´t be described through geometrical optics. Instead, the proper explanation is given through the behavior of the Guoy phase and the Uncertainty Principle (explanation), both characteristics that without been “difficult” in the mathematical sense, are quite complicated to understand "intuitively" even for people with graduate level of engineering knowledge (at least for me and my classmates as sample). But let think about it from a different angle: since monochromatic light, as a laser light basic model, is bandlimited, the rate of change of any part of the electromagnetic field must be bounded, then, since each differential section of the propagation plane could be approximated by plane wave $A(x,y)\,e^{\vec{x}\cdot\vec{k}-ct}$, I believe that a more intuitive explanation could be that because near the waist of the beam the electromagnetic profile have gained so much curvature, that its shared rate of change between the time and spatial components are reaching this "maximum rate of change restriction", and since the time characteristics is unaltered in the lineal model of the wave equation, and the differential portion of the beam cannot gain more curvature, the energy doesn´t have more alternative than change its direction given by the cross-product $\vec{x}\cdot\vec{k}$ - Caution here because this is not a "real" explanation, but something I am trying to prove but for which I need to find these $\text{mysterious conditions}\,\mathbb{X}$, or even more, if right, maybe is possible to design a Graded Index Lens to "smooth" these rate of change transitions and achieve smaller spot sizes, with a direct application in microscopy.
Now being full speculative, my guts also tell me that it could be also helpful to give more intuitive explanation to things like where comes from the Stationary Action Principle, The Second Law of Thermodynamics, The Ultraviolet Catastrophe, Causality, and many other things were the change of rate or Fourier transforms are involved. But at least, having the upper bound could be useful to check if spurious rate of change is being introduced (or being left out), because of using functions that aren´t compact-supported. Maybe these things are obvious for physicists, but for me as an engineer they wasn´t at all, so for reaching so far, thanks you very much.
