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I'm interested in linear combinations of cosines: $$f(x) = \alpha_1 \cos(2\pi \theta_1 x) + \alpha_2 \cos(2\pi \theta_2 x) + \cdots + \alpha_k \cos(2\pi \theta_k x)\enspace,$$ where $\alpha_i \in \mathbb{Z}$ and the $\theta_i$'s are linearly dependent irrational numbers (when they are rational numbers or irrational linearly independent, I have less trouble, as in the former case the function is periodic, and in the second, any value of each $\cos$ can be attained independently of the others).

When plotting functions like this, one sees that they are not periodic, but still, they are close to be. In particular, it seems that there are values $p$ and $\epsilon$ such that if $f(x) > \epsilon$, then $f(x+n\times p) > \epsilon$, at least for the first few $n$'s. I have two questions:

  • Is this true?
  • Is there a name for the pseudo-periodicity that $f$ seems to enjoy?


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They are almost-periodic functions. – Henning Makholm Jun 28 '12 at 20:24
up vote 1 down vote accepted

You could be interested in almost periodic functions.

A function $f\colon \Bbb R\to \Bbb C$ is said almost periodic if for all $\varepsilon>0$, the set $$P(f,\varepsilon)=\{T\in\Bbb R\mid\forall x\in \Bbb R, |f(x+T)-f(x)|\leq \varepsilon\}$$ is relatively dense, that is, we can find $L>0$ such that each interval of length $L$ has a non-empty intersection with $P(f,\varepsilon)$.

It can be shown that if $f\colon\Bbb R\to\Bbb C$, the following properties are equivalent:

  1. $f$ is almost periodic;
  2. for all $\varepsilon>0$, we can find an integer $N$, real numbers $\theta_1,\dots,\theta_N$ and complex numbers $a_1,\dots,a_N$ such that $$\forall x\in\Bbb R,\quad \left|f(x)-\sum_{j=1}^Na_je^{i\theta_j x}\right|\leq \varepsilon.$$
  3. The family $\{\tau_t f,t\in\Bbb R\}$ has a compact closure for the uniform norm on the continuous bounded functions over $\Bbb R$, where $\tau_tf(x)=f(x+t)$.

This concept has been introduced by Harald Bohr. A good reference is Corduneanu's book Almost periodic functions.

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