Questions tagged [almost-periodic-function]

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Can we calculate the probability that $f(x)$ is positive for a randomly chosen value of $x\in(0,m)$ as $m\to\infty$? (uniform distribution)

Following my previous question here, I have this function $$f(x)=10+3 \cos (ax-bx)+13 \cos (ax+bx)+2 \cos (\frac32 a x)+17 \cos (b x),$$ with $\frac ab \notin \mathbb{Q}$. What is the limit $$ \lim_{m\...
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1answer
208 views

Does such a function exist?

I am looking for a function with the following property: Let $v_1,v_2$ be two linearly independent vectors in $\mathbb{R}^2.$ I am given a smooth function $g:(0,1) \rightarrow (0,\infty).$ I am trying ...
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1answer
62 views

On $B^1$ and $B^2$ almost-periodic functions

The Besicovitch class of $B^p$ almost-periodic functions is defined as the closure of the set of trigonometric polynomials (of the form $t \mapsto \sum_{n=1}^N a_n e^{i \lambda_n t}$ with $\lambda_1, \...
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111 views

Aperiodic packings of the plane with disks of multiple radii

Does there exist a finite set of radii such that some aperiodic packing of the plane by disks of those radii is believed to achieve the maximal packing density (not achieved by any periodic packing)? ...
3
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1answer
283 views

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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1answer
93 views

Long time average of solution to ODE with almost periodic structure

I encountered the following question in my studies: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a Bohr almost periodic function such that $\inf_{\mathbb{R}} f = 0$ but $f(x) > 0$ for all $x\in \...
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330 views

Can we characterize a periodic function by the compactness of the set of its translates?

Given a function $f$, let us define the translates $f_t(x)=f(x-t)$. A (Bochner) almost-periodic function is a bounded continuous function on $\mathbb R^\nu$ such that the set of functions $\{f_t\vert ...
3
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1answer
241 views

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 ...
3
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1answer
185 views

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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0answers
38 views

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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2answers
666 views

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 ...
6
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2answers
2k views

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 ...
2
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1answer
217 views

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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0answers
73 views

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}}\,...
2
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1answer
93 views

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 ...
2
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1answer
165 views

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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47 views

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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1answer
164 views

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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1answer
177 views

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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1answer
691 views

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 ...
5
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1answer
328 views

Discrete spectrum and almost periodicity

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 \...
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1answer
398 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 ...
5
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1answer
640 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 ...
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285 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{...
4
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1answer
370 views

Using a quadratic kernel instead of a linear kernel in the Laplace transform

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) \: ...