A function is square integrable when the integral of its square is finite.

A given function f(x) is "square integrable" if its "L2 norm" is finite (note the capital L, as distinct from the l2 norm used on a vector). The exact representation of the L2 norm is commonly written as follows:

 / ∞
\         2
 \  |f(x)|  dx
/ -∞


A square integrable function f(x) falls under the Titchmarsh theorem, which relates Fourier transforms to both the boundedness of f(x) and Hilbert transforms.

In statistics, the variance of a function is related to its L2 norm: a given function has a defined variance iff it is square integrable.

Thanks to ariels for the reminder about the relationship to probabilities, and the simple fact that square integrability is an underpinning of many other things including being the first example of a Hilbert space.

Examples of square integrable functions

* Any probability distribution function which has a variance, e.g., normal distribution, poisson distribution (only in its piecewise form), gamma distribution, etc.
* Any piecewise function which is zero everywhere except a finite number of finite intervals which are all bounded, e.g., f(x) = x for x in (7, 20), 0 otherwise.

One Hilbert space in particular is "the set of all functions in R x R which are square integrable." This space also happens to be a Banach space (by consequence of the fact that every Hilbert space is also a Banach space).

Information gathered from the following locations:

Eric W. Weisstein. "Square Integrable." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SquareIntegrable.html
Eric W. Weisstein. "L2-Norm." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/L2-Norm.html
Eric W. Weisstein. "Titchmarsh Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TitchmarshTheorem.html
Eric W. Weisstein. "Hilbert Space." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HilbertSpace.html

So far nobody mentioned the most important application of square integrable functions. Well, the most important one I know of at least ...

Wave functions in quantum mechanics have to be square integrable.

That's because the probability of a particle being observed at a given point in space is the square of the value of the wave function at that point. And of course the integral over all points must give a probability of one - the particle has to be somewhere after all. Therefore we have to normalize a potential wave function so that the integral over the squared function is one, and that can only work if the integral is finite.

This has some consequences, most importantly all wave functions that do not approach zero for x->∞ may be dismissed straight away as unphysical. On the other hand, the physicists' most beloved wave functions, namely plane waves, are not square integrable either. A plane wave has a sharp momentum and because of the uncertainity relation the spatial probability density is constant everywhere. But as long as one keeps in mind that they don't actually exist it's ok to use them :)

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