Almost periodic functions

When you see the word almost" in a mathematical context, it might be used informally, but often it has a precise meaning. I wrote about this before in the post Common words that have a technical meaning.

Often the technical meaning of almost" is within any finite tolerance." That's how it is used in the context of almost periodic functions.

A function f is periodic with period T if for all x,

f(x + T) = f(x)

For example, the sine function is periodic with period 2.

So what does almost periodic mean? It means that for any > 0, there exists a T > 0 such that

|f(x + T) - f(x)| <

for all x. Note that the value of T depends on the value of . You tell me your tolerance for almost" and in theory I could hand you back T that meets your tolerance.

That's in theory. Can we actually do it in practice? Let's consider

f(t) = sin(2t) + sin(2t)

where / is irrational. In the previous post we had = 1/log 2 and = 1/log 5.

The Hurwitz approximation theorem says that because / is irrational, there are infinitely many integers p and q such that

|/ - p/q| < 1/(5q^2).

To put it another way, we can find p and q such that

|q - p| < /5q,

i.e. we can find p and q such that q and p are close together as we wish by looking for a large enough q, and Hurwitz promises us that we can find a q as large as we want.

If we let T = q/ then the first component of f is exactly periodic, i.e.

sin(2(t + q/) = sin(2t).

With a little effort we can show that

sin(2(t + q/)) = sin(2t + 2(q - p)t/)

and we said above we can make

q - p

as small as we like by choosing a large enough q in Hurwitz' theorem. And so we can choose q large enough that

sin(2(t + q/)) - sin(2t)

is uniformly as small as we'd like.

The post Almost periodic functions first appeared on John D. Cook.