# Questions tagged [almost-periodic-function]

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### Almost periodicity of Bessel functions

We know that a periodic function (e.g. a trigonometric function) has the property $$f(x+n\Lambda)=f(x) \qquad n\in\mathbb Z$$ A Bessel function is not exactly periodic, because the value of the ...
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### What are the almost periodic functions on the complex plane?

The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally ...
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### About understanding manifold structure on WAP compactification of $\Bbb{C} \rtimes \Bbb{T}$

Let $G$ be a locally compact topological group. A continuous bounded function $f$ on $G$ is called (weakly) almost periodic if the set $L_Gf$ of left translates is relatively compact in the (weak) ...
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### Do maximally almost periodic groups embed homeomorphically into their Bohr compactifications? [duplicate]

If $G$ is a Hausdorff topological group and $bG$ is its Bohr compactification, Wikipedia says that $G$ is called maximally almost periodic (MAP) if and only if the natural map $i : G \to bG$ is ...
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### Bohr compactification as a topological compactification

Let $G$ be a locally compact Hausdorff group. Denote its Bohr compactification by $bG$. Despite group structure, $G$ has several (Hausdorff) compactifications that, in a sense, the smallest one is ...
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### Distribution of distances of successive zeros of $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$

Let $\alpha$ and $\beta$ be incommensurate real numbers. Consider the function $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$ and its positive zeros $x_k(\alpha,\beta)$. Fix $\alpha$ and ...
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### Approximation of quasi-periodic function by trigonometric polynomials

The elements of the closure of $\{ \sum_{j=1}^n a_j e^{i\nu_j x}: a_j\in \mathbb{C}, \nu_j\in \mathbb{R} \}$ in the supremum-norm are called almost periodic functions. An almost periodic function $f$ ...
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### Riesz transform on almost periodic functions?

It is well established that the Riesz transform is well-defined for $f\in L^p(\mathbb{R}^d)$ via \mathcal{R}_jf(x) = c_d\lim_{\epsilon\to 0}\int_{|x-y|>\epsilon}\frac{(x^j-y^j)f(y)}{|x-y|^{d+1}}\,...
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### Distribution of an almost periodic trigonometric polynomial

Consider an almost periodic trigonometric polynomial $f(t)=e^{i2\pi t} + e^{i 2\pi \lambda t}$ for some irrational $\lambda$. I'm interested in distribution of such polynomial. In other words, is ...
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### Dominated convergence for quasi-periodic functions

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 ...
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### Fourier exponents of an almost periodic solution of differential equation

Consider an almost periodic solution $u$ of some differential equation (autonomous, nonautonomous, ordinary or in partial derivatives). There are methods to show the existence and uniqueness of such ...
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### Is $f(n)=\cos(2\pi\theta(1+2+\ldots+n))$ almost periodic?

We use the following definition of almost periodicity given in https://arxiv.org/pdf/math-ph/0005018.pdf Given a bounded function $f :\mathbb Z \to\mathbb R$ we denote the set of translates of $f$ by ...
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### Almost periodic functions and the property A

A subset $A \subset \mathbb{R}$ is relative dense if there exists a real number $L>0$ such that for every $t \in \mathbb{R}$ the set $A \cap [t,t+L]$ is not empty. Such number $L$ is called an ...
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### What does the unique mean on weakly almost periodic functions look like?

There is a unique invariant mean $m$ on WAP functions on any discrete group (see definitions below, theorem of ?). However, the proofs I found are fairly non-explicit on how to obtain this invariant ...
According to Vershik, an ergodic invertible measure-preserving transformation $T$ on a Lebesgue space $X$ has discrete spectrum if and only if for every bounded measurable function $f\colon X \to \... 1answer 390 views ### C* algebras of Almost Periodic Functions Suppose we take, for example, the$C^*$-algebra which is the sup norm closure of the exponentials$e^{2 \pi i ax}$where$a \in \mathbb{Z} + \theta \mathbb{Z}$for$\theta$an irrational number. This ... 1answer 575 views ### Besicovitch Almost Periodic Functions a subspace of what? The common example of a nonseparable Hilbert space comes from the collection of Besicovitch almost periodic function spaces. Starting with$L^p_{\text{loc}}(\mathbb{R})$we look at those elements ... 0answers 274 views ### Non-crystallographic cluster algebras Background Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed$(\mathbf{...
Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that \$\int_0^\infty f(t) \exp(-xt^2) \: ...