# Questions tagged [almost-periodic-function]

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25
questions

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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\...

**5**

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**1**answer

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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**1**answer

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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**0**answers

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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**1**answer

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 ...

**2**

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**1**answer

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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**1**answer

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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**1**answer

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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**1**answer

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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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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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 ...

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**2**answers

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**

votes

**1**answer

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 $f$ ...

**1**

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**0**answers

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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**1**answer

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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**1**answer

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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**1**answer

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 ...

**3**

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**1**answer

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 ...

**7**

votes

**1**answer

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 ...

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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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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 ...

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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{...

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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) \: ...