Questions tagged [almost-periodic-function]

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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 ...
Darren Ong's user avatar
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1 answer
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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 ...
mkreisel's user avatar
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7 votes
2 answers
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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 ...
ARG's user avatar
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6 votes
2 answers
991 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 ...
XIII's user avatar
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2 answers
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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 ...
Heis 's user avatar
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1 answer
419 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 \...
Stéphane Laurent's user avatar
6 votes
0 answers
119 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)? ...
James Propp's user avatar
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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{...
Philipp Lampe's user avatar
5 votes
1 answer
228 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 ...
Sascha's user avatar
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1 answer
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Intuition for almost periodic solution and Poincaré recurrence theorem

I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer. Suppose that we have a PDE that admit a solution $u$ that can be ...
Niser's user avatar
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A limit related to quasi-periodic function

Let us consider $V(x) = 2-\sin(x) - \sin(\sqrt{2} x)$ on $x\in \mathbb{R}$ so that $V(x)>0$ everywhere. One can see that $$ \frac{C_1}{t^2} \leq \min_{|x|\leq t} V(x)\leq \frac{C_2}{t^2} $$ ...
Sean's user avatar
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1 answer
788 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 ...
Greg Zitelli's user avatar
4 votes
1 answer
394 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) \: ...
James Propp's user avatar
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0 answers
106 views

Quasi-crystaline generalization of elliptic functions

I came across some meromorphic function, call it $f(z)$, which is "quasicrystalline" in the following sense: one can write $f$ as: $$ f(z)=\frac{\sum_i a_i e^{i(q_{i,x}x+q_{i,y}y)}}{\sum_i ...
Yarden Sheffer's user avatar
3 votes
1 answer
322 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 ...
Merry's user avatar
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1 answer
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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 ...
demolishka's user avatar
3 votes
1 answer
780 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 ...
DrManhattan's user avatar
3 votes
1 answer
207 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) ...
Mambo's user avatar
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1 answer
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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 ...
Darren Ong's user avatar
2 votes
1 answer
139 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 \...
Sean's user avatar
  • 313
2 votes
1 answer
311 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 $f$ ...
Severin Schraven's user avatar
2 votes
1 answer
114 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 ...
demolishka's user avatar
2 votes
1 answer
242 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 ...
Severin Schraven's user avatar
2 votes
0 answers
56 views

Rotation number for multicomponent Schrödinger equation

Rotation number for Schrödinger equation of the form \begin{equation} -x''(t) +q(t) x(t) = E x(t) \end{equation} was defined in R. Johnson J. Moser "The rotation number for almost periodic ...
0x11111's user avatar
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2 votes
0 answers
53 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 ...
demolishka's user avatar
1 vote
3 answers
235 views

Can we calculate the probability that $f(x)$ is positive for a random $x\in(0,m)$ 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\...
user avatar
1 vote
1 answer
98 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, \...
A. Bailleul's user avatar
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1 vote
1 answer
101 views

Bounds of periodic functions formed from infinite series of shifts

Recently, I have become quite obsessed with the follow series: $$f(t,x)=\sum_{m=-\infty}^{\infty} (-1)^m f(t+x m)$$ where $f$ is analytic. This series automatically produces a periodic function with ...
Bobby Ocean's user avatar
1 vote
0 answers
41 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 ...
Alex M.'s user avatar
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1 vote
0 answers
89 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}}\,...
Xuxu's user avatar
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0 votes
0 answers
38 views

Lower bound of infinite sum of shifts

This is an extension of https://mathoverflow.net/posts/452526. So, it appears that $\sin(xt)$ serves as a better lower bound instead of a linear equation, in fact the series appears to uniformly ...
Bobby Ocean's user avatar