I am doing some work regarding quasiperiodic functions but I am not able to figure out the difference between almost periodic and quasiperiodic functions. Can anyone let me know about it?

  • $\begingroup$ there are many possible definitions, all different of quasi or almost periodicity. what are yours ? $\endgroup$ – reuns Apr 5 '16 at 4:52
  • $\begingroup$ For almost periodic functions I am considering H. Bohr definition. I didn't find any statement or definition which clarify the difference between them. $\endgroup$ – ashu sharma Apr 5 '16 at 5:00
  • $\begingroup$ I'm asking you to write down of give some links to the definitions you are considering $\endgroup$ – reuns Apr 5 '16 at 5:09
  • $\begingroup$ and on wikipedia, almost-periodic is : for every $\epsilon >0$ there is $T(\epsilon) >0$ such that for every $t$ : $f|(t+T(\epsilon))-f(t)| < \epsilon$, while quasi-periodic is : there is a $\epsilon$ relatively small and a $T > 0$ such that for every $t$ : $f|(t+T)-f(t)| < \epsilon$. $\endgroup$ – reuns Apr 5 '16 at 5:19

I am just a master student writing a thesis in that direction, but maybe you find it helpful nonetheless.

One can show that the Bohr almost-periodic functions are the closure of the trigonometric functions in the supremum norm, i.e.

$$ \mathcal{A}:= \overline{\{ \sum_{1\leq j \leq n} a_i e^{i \nu_j x} : n \in \mathbb{N}, a_i\in \mathbb{R}\}}^{(C_b(\mathbb{R}, \mathbb{C}), \Vert \cdot\Vert_{sup})} .$$

Intuitively the difference is that the "Fourier series of a quasi-periodic function contains less independent frequencies than a general almost-periodic function". On the space of Bohr almost-periodic function we have the following sesquilinear form:

$$ \langle f, g\rangle = \lim_{x\rightarrow \infty} \frac{1}{x} \int_{0}^x f(t)\overline{g(t)}dt. $$

The frequency module $M(f)$ (the $\mathbb{Z}$-module of all frequencies that may appear in the formal Fourier series of $f$) is defined the $\mathbb{Z}$-module generated by

$$ \{ \nu\in \mathbb{R} : \langle f, e^{i\nu x}\rangle\neq 0\}.$$

We call $f$ quasi-periodic if its frequency module is finitely generated over $\mathbb{Z}$. For example, a function is periodic iff its frequency module is generated by single frequency.

There is an alternative characterization of quasi-periodic functions. Namely, $f$ is quasi-periodic if there exist a continuous map $Q:\mathbb{T}^n \rightarrow \mathbb{C}$ and a "frequency vector" $\omega=(\omega_j)_{j=1}^n$ such that $f(x)=Q(x\cdot \omega)=Q(x\omega_1, \dots, x\cdot \omega_n)$. Hence, the motion of the quasi-periodic function "lives on a finite dimensional torus". E.g.

$$ f(x)= \sin\left(\frac{2}{7}2\pi x\right) + \sin(\sqrt{2}\cdot 2\pi x) $$

has frequency module $\{ \frac{7}{2}k + \frac{1}{\sqrt{2}}l : k, l \in \mathbb{Z} \}$ and "lives on a 2-dimensional torus". Where a general almost-periodic function can be thought of "living on an infinite-dimension torus".

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  • $\begingroup$ @ Severin Thanks $\endgroup$ – ashu sharma Apr 17 '16 at 4:24
  • $\begingroup$ @ Severin I still have some doubts. Is the frequency module of almost periodic function contains finite frequencies or infinite frequencies.And are these frequencies rationally independent? $\endgroup$ – ashu sharma Apr 17 '16 at 4:30
  • $\begingroup$ The frequency module is (except for constant functions) never finite (see below). But for for quasi-periodic functions it is finitely generated over $\mathbb{Z}$, i.e. there are only finitely many rationally independent frequencies. You cannot hope the frequency module to be finite. This does not even work in the periodic setting. If $\omega$ is a frequency then $n \omega$ is a frequency as well for every $n\in \mathbb{N}$, e.g. $1$-periodic functions are $2$ periodic as well. $\endgroup$ – Severin Schraven Apr 17 '16 at 9:46

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