Mathematics

Questions tagged [almost-periodic-functions]

Use this tag for questions related to almost periodic functions, which are functions of a real number that are periodic to within any desired level of accuracy given suitably long, well-distributed "almost-periods".

An almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy given suitably long, well-distributed, "almost periods." There is also a notion of almost periodic functions on locally compact abelian groups.

Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. An example would be a system of planets with orbital periods that are not commensurable, i.e., with a period vector that is not proportional to a vector of integers. A theorem of Kronecker from Diophantine approximation can be used to show that any particular configuration that occurs once will recur to within any specified accuracy: if we wait long enough we can observe the planets return to within a second of an arc to the positions in which they once were.

57 questions
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Minimum values of the sequence $\{n\sqrt{2}\}$

I have been studying the sequence $$\{n\sqrt{2}\}$$ where $\{x\}:= x-\lfloor x\rfloor$ is the "fractional part" function. I am particularly interested in the values of $n$ for which $\{n\sqrt{2}\}$ has an extremely small value - that is, when…
Franklin Pezzuti Dyer
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Average Distance Between Zeroes of $\sin(x)+\sin(x\sqrt{2})+\sin(x\sqrt{3})$

QUESTION: What is the average distance between the consecutive real zeroes of the function $$f(x)=\sin(x)+\sin(x\sqrt{2})+\sin(x\sqrt{3})$$ or, more specifically, if $z(x)$ is defined as the number of zeroes $\zeta$ satisfying $|\zeta|<x$ , what is…
Franklin Pezzuti Dyer
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What is the Haar measure on the Bohr compactification $b\mathbb{Z}$ of the integers?

What is the Haar measure on the Bohr compactification $b\mathbb{Z}$ of the integers? We (my collaborators and I) suspect that it is $$ \int_{b\mathbb{Z}} d\mu_H f(n) = \lim_{N\to\infty}\frac{1}{2N+1} \sum_{n=-N}^{N} f(n) ~. $$ This is finite and…
dac
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Why Bohr's definition of almost periodicity?

Bohr in his Almost Periodic Functions book writes that the following naive approach to almost periodicity does not work, he phrases it being not a happy choice . Specifically, let $\mathcal A$ denote the functions defined over $-\infty < x <…
hyportnex
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Symmetry in an almost periodic function

A comment under this answer suggests looking at the graph of $$f(t) = \sin t + \sin(\sqrt 2\ t) + \sin(\sqrt3\ t),$$ and I did so, on the interval $0\le t\le 60.$ I was struck by a seeming near-symmetry, so I let $$g(t) = f(60-t)$$ and superimposed…
Michael Hardy
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An example of almost periodic function

" I need a continuous almost periodic function $f(x)$ such that $\lim_{x\to\infty}f(x)$ exists. But this function should not be constant, which is a trivial example ." Definition of almost periodic function:…
guest
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"Beats" via trig identity or something?

Planning on talking about resonance in DE. The solution to the IVP $$y''+y=\cos(t),\quad y(0)=y'(0)=0$$is $$y=\frac12 t\sin(t).$$Resonance, great. Now what if the forcing function has almost the resonant frequency? If $\alpha^2\ne1$ the solution to…
David C. Ullrich
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Is there any research about a function with changing "period" like sin(1/x)?

I'm encountering a function with a "changing period". It has some sense of period but not exactly. For example: f(x) = cos(1/x). Generally, it has the form of f(x + g(x)) = f(x). I cannot find any information about it. Is there any research about…
Fengfeng
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Almost periodic function

I am trying to do this problem but could figure it out. Show that $f(x) = \cos 2\pi x +\cos 2\pi \sqrt{2} x$ is almost-periodic by showing directly that given $\varepsilon > 0$ there exists an integer $M$ such that at least one of any $M$…
user93736
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Can the definition of almost periodic functions be simplified?

A bounded continuous function $f : \mathbb{R} \to \mathbb{C}$ is almost periodic if for every $\epsilon>0$ , there exists some $L>0$ , such that every interval of $\mathbb{R}$ with length $\ge L$ contains some real number $T$ such that $\Vert…
Christophe Leuridan
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Function $z(\theta) = e^{i \theta} + e^{i \theta \pi}$ is not periodic because $\pi$ is irrational.

There is this popular video going around that shows $\pi$ is irrational using visualization of the function, $z(\theta) = e^{i \theta} + e^{i \theta \pi}$ . I understand the reason intuitively, both the functions have different periodicities so the…
Naren Nallapareddy
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Intuition for almost periodic solution and Poincaré recurrence theorem

Suppose that we have a PDE that admit a solution $u$ that can be expressed in a certain system of coordinates (angle-action variables) as advection with constant velocity on tori. And suppose also that the solution is almost periodic i.e. $$\forall…
Niser
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Which functions do Dirichlet series represent?

I'm reading Serre's $\textit{A course in Arithmetic}$ where he defines a Dirichlet series to be an infinite sum of the form $$f(z) = \sum\limits_{n=1}^{\infty} a_ne^{-\lambda_nz} $$ where $\lambda_n$ is an increasing sequence of reals diverging to…
Zzzzz
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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 $x\in \mathbb{R}$ . An example is $$ f(x) = 2-\sin(2\pi x) -…
user93736
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A counterexample to convergence in $B^2$ implies convergence in $L^2_\text{loc}$

In regards to this question , I feel I can produce a complicated counterexample as follows. I wonder if I have made a mistake in this argument. We know that $m+n\sqrt{2}$ is dense in $\mathbb{R}$ as $m,n$ vary over the integers. Therefore, let…
Thangachelli Debopritama
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