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I was reading through computer networks, and I saw an expression that I had never seen before when discussing Fourier transformations: “reasonably behaved function.”

What does this mean about a function? I am not able to find an explanation online. Is this expression commonly used?

EDIT: The actual question used in the book:

In the early 19th century, the French mathematician Jean-Baptiste Fourier proved that any reasonably behaved periodic function, $g(t)$ with period $T$ can be constructed as the sum of a (possibly infinite) number of sines and cosines.

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  • $\begingroup$ Possible explanation here. $\endgroup$ Sep 5, 2017 at 12:39
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    $\begingroup$ It depends on finer context; good properties include continuity, existence of classical derivatives, integrability, and integrability of weak derivatives, among others. $\endgroup$
    – Ian
    Sep 5, 2017 at 12:39
  • $\begingroup$ Write down the exact sentence, it's difficult to say something without a context. $\endgroup$
    – Blex
    Sep 5, 2017 at 12:40
  • $\begingroup$ This is my take, although I have nothing concrete to back it up: It is not an explicitly defined notion, but there are some pretty weird functions out there, and I am sure the author just wanted to rule all those out. He is hijacking the reader's intuition about how nice a funciton "ought" to be in order to not have to come up with actual, concrete requirements that will have to be backed up and are possibly too strong or too weak. Also, such a discussion would distract from what he is actually trying to teach you: Fourier transformations. $\endgroup$
    – Arthur
    Sep 5, 2017 at 12:40
  • $\begingroup$ Look up the phrases "regularity of a function", "well-behaved function", "smooth function" to get a feel of what's at stake. $\endgroup$ Sep 5, 2017 at 12:40

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As a rule of thumb, I would say that when a computer scientist or a physicist says "well behaved function" they mean precisely this: "Any function for which the mathematical theorem I am about to quote is true, only I don't remember exactly which functions these are".

Disciplines that use mathematics, such as computer science and physics, are heavily relying - of course - on mathematics, but their presentations often gloss over the technical details which make mathematics precise. This is especially omnipresent in the theory of Fourier transforms, where many theorems have to be carefully formulated in terms of the functions involved. Thus, I heard a professor at MIT, in a lecture on Quantum Mechanics, say that the Fourier transform is invertible for any function, and then - almost apologetically - added that mathematicians are more careful in formulating the theorem.

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This is not a well-defined term, although it is a common one. The phrase "reasonably behaved function," or its many variants, could mean any number of things. Sometimes it means continuous or some similar condition.

Often in mathematics, but especially in applications of mathematics such as physics, we expect or know something is true, but the precise conditions under which it is true are complex, unimportant, or uninteresting. That is part of the reason this terminology exists. Of course, if I write "$P$ is true for any reasonably behaved function," then there ought to exist precise conditions a function must meet for $P$ to be true.

As an example, there are lots of weird things that can happen in general topological spaces that do not happen in reasonably well-behaved spaces. What exactly this means is unclear. A space being Hausdorff may be a necessary and sufficient condition for some weird thing not to happen, for example. Sometimes the exact condition will be clear to the reader. Sometimes the author is just being a bit lazy, but more often the idea is that the details are distracting or unnecessary; they can confuse the reader by putting their intuitions into question, which is not always ideal for understanding, even if addressing the technicalities eventually is a good idea. For a more specific example, note that $a+b=b+a$ only for reasonably behaved algebraic systems such as abelian groups. (The groups $S_n$ are only abelian for $n \leq 2$.)

The opposite of a well-behaved object is a pathological object. Pathologies are really nasty or weird things, like an everywhere continuous but nowhere differentiable function or the Cantor set. Looking at the Wikipedia page for Pathological (Mathematics) may give you a better idea of what this terminology means. I implore you look at the examples for guidance.

To address the particular quote in question, the author is talking about Fourier series. To have an exact construction, differentiability will do, although that is not the weakest condition for which the construction is exact (i.e., the Fourier series converges everywhere). Also, oftentimes convergence almost everywhere (in the technical sense) is OK, or one may want more than just pointwise convergence (e.g., uniform convergence), and in these cases we may require something stronger or weaker than differentiability.

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It depends on the context of the phrase. It might mean, for instance, that the function is smooth. It generally means that the function is not pathological in any way, so is not, say, a special case with some odd behaviour.

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