A function $f:\mathbb{R} \rightarrow \mathbb{C}$ is called almost periodic if it is the uniform limit of trigonometric polynomials. One can show that for almost periodic $f$ the following pointwise limit (called the mean value of $f$) exists
$$ M(f) = \lim\_{x\rightarrow \infty} \frac{1}{x} \int\_0^x f(t)dt.$$
We call a almost periodic function $f$ quasi-periodic if the $\mathbb{Z}$-module
$$ \mathcal{M}(f)=\langle \nu\in \mathbb{R} : M(f\cdot e^{-i \nu x}) \neq 0 \rangle\_{\mathbb{Z}} $$
(called the frequency module of $f$) is finitely generated. Further we set
$$ \mathcal{A}(f)=\{ \text{ g quasi-periodic and }\mathcal{M}(g) \subseteq \mathcal{M}(f)\}.$$



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> My question is whether there are known results relating pointwise limit of quasi-periodic functions and the convergence of the associated mean values.
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For example is there a version of Lebesgue's dominated convergence theorem for $M(\cdot)$? I think the minimal assumptions should look something like this


**Proposition:** Let $f$ be a quasi-periodic function and let $(f\_n)\_{n\in \mathbb{N}}\subseteq \mathcal{A}(f)$ be a sequence bounded in the supremum norm and $f\_n \rightarrow f$ pointwise then $M(f\_n)\rightarrow M(f)$.


Any reference would be greatly appreciated.


**Edit:** For me a trigonometric polynomial is a function of the form


$$ \sum\_{j=1}^n a\_n e^{i \nu\_j x}, $$


where $n\in \mathbb{N}, a\_j\in \mathbb{C}$ and $\nu\_j\in \mathbb{R}$.



Dominated convergence for quasi-periodic functions
The proposition is indeed true. To see this one uses an alternative characterization of quasi-periodic functions. Let $\omega\_1, \dots, \omega\_m\in \mathbb{R}$ a $\mathbb{Z}$-basis of the frequency module of $f$ and set $\omega= (\omega\_1, \dots, \omega\_m)$. By the work of Harald Bohr, we have that for every $f\_n\in \mathcal{M}(f)$ exists a unique $F\_n\in C^0(\mathbb{T}^m, \mathbb{C})$ such that


$$ f\_n(x) = F\_n(x\omega). $$


Furthermore, one can prove that


$$M(f\_n) = \frac{1}{(2\pi)^m} \int\_{\mathbb{T}^m} F\_n(\theta)d\theta.$$


As $\{ x\omega \in \mathbb{T}^m: x\in \mathbb{R}\}$ lies dense in $\mathbb{T}^m$, pointwise convergence $f\_n \rightarrow f$ implies pointwise convergence of $F\_n \rightarrow F$ on the entire torus. Furthermore, the uniform boundedness of the $f\_n$ carries over to the uniform boundedness of the $F\_n$. Hence, the proposition follows by dominated convergence.


