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".

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Parameters Estimation of LTI model using Single Period Response Segment to Quasi-Periodic Signal

I have problem of parameters estimation LTI system using single period fraction of response signal to quasi-periodic input signal, with known single period shape pattern and period times. System ...
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28 views

A strong version of uniform continuity for almost periodic functions

Let AP be the space of almost periodic functions on $\mathbb R$ (defined as the uniform closure of the set of generalized trigonometric polynomials $\sum_{n=1}^m a_n e^{i\lambda_n t}$, where $a_n\in \...
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114 views

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 $...
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32 views

How to map “smooth hills” to “bumpy” hills?

Assume I have some function $f$ such that its range is $[0,1]$ Assume the function looks somewhat like this: (and excuse the lack of artistic skill) : (Note the function is NOT actually periodic, i....
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51 views

Convergence in the topology of $B^2$ implies convergence in $L^2_{loc}(\mathbb{R}^d)$

Regarding the question here, I wanted to ask whether the statement is true the other way round. Let $f_n$ be a sequence of functions in the Besicovitch space of almost periodic functions $B^2(\...
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338 views

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 ...
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50 views

Period of a particular Cycle for a Bessel Function

The Bessel Function of the First Kind $J_a(x)$, and the Bessel Function of the Second Kind $Y_a(x)$, at least when $a$, is an integer or half integer are cyclical, as their values go from positive to ...
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$e^{\tau \sqrt{163*4}} \approx 475\dots792$ is almost an integer

$$ \frac{log(4750778730825177725463920948909726618214491718039471366318747406368792)}{ sqrt(652) } - \tau = -2.54282421310320265217436545223140117387 E-78 $$ found with Pari GP by playing with $e^{\...
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44 views

Quasi-periodic sequence

Let $f(\theta)$ be some $2\pi$-periodic function which takes the values $f(\theta) \in \{1,-1\}$. Further let $Q$ be some number which is rationally independent of $2\pi$ (More specifically take $Q/(2\...
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74 views

Convergence in the topology of $L^2_\text{loc}$ implies convergence in $B^2$?

Let $f_n$ be a sequence of functions in $L^2_\text{loc}(\mathbb{R})$ which converge to a function $f\in L^2_\text{loc}(\mathbb{R})$ in the topology of $L^2_\text{loc}(\mathbb{R})$, i.e., $f_n\to f$ in ...
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1answer
53 views

Mean value of an almost periodic function

Consider the set of all trigonometrical polynomials of the form $P(x)=\sum_{j=1}^n a_n e^{ix\cdot\xi_n}$, where $\xi_n\in\mathbb{R}^d$. A function is said to be almost periodic in the sense of Bohr if ...
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86 views

Besicovitch almost periodic functions with seminorm zero

Consider the set of all trigonometrical polynomials of the form $P(x)=\sum_{j=1}^n a_n e^{ix\cdot\xi_n}$, where $\xi_n\in\mathbb{R}^d$. A function is said to be almost periodic in the sense of Bohr if ...
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382 views

Is Heaviside step function or unit step function periodic?

I have a unit(or Heaviside) step function in discrete form: $$\text u[n]=\begin{cases} 0, & n < 0, \\1, & n \ge 0, \end{cases}$$ and in continuous form: $$\text u(t)=\begin{cases} 0, &...
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Invertibility of derivation on $AP(\mathbb{R},\mathbb{R}^n)$

I have following question based on this paper: https://boundaryvalueproblems.springeropen.com/track/pdf/10.1186/s13661-016-0576-9 The author defines the function spaces $AP(\mathbb{R},\mathbb{R}^n)=...
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59 views

Special function $ 1 + \sum_{n=1}^{\infty} n \frac {z^{t(n)}}{t(n)!} $?

Consider the “ special function “ $$ f(z) = 1 + \sum_{n=1}^{\infty} n \frac {z^{t(n)}}{t(n)!} $$ Where $t(n)$ means n th triangular number. I assume it has No closed form despite resembling exp ...
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878 views

what is the fundamental period of $\sin{x}$ and $\sin[x]$

The question was which of the following is a periodic function, in which two of the possible answers which I thought of were $\sin{\{x\}}$ and $\sin[x]$ where $\{.\}$ and $[.]$ are fractional part and ...
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223 views

Showing integral of continuous function is differentiable

We define $F(x)$ = $\frac{1}{2a}$$\int_{-a}^{a} f(x+t) dt$, for some $f(x)$ which is continuous on R. I need to show that $F(x)$ is differentiable and has a continuous derivative. I am having trouble ...
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167 views

Almost periodic function with mean value zero

I want to prove the following (I believe it is true but I am not sure) If $f\in \mathrm{AP}(\mathbb{R})$, where the space of almost periodic functions $\mathrm{AP}(\mathbb{R})$ is defined in the ...
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86 views

“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 ...
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34 views

Period of f given [duplicate]

So the problem states that if $f(z)$ is entire, and satisfies the relation $f(z+i) = f(z)$ and $f(z+1) = f(z)$, show that $f(z)$ is constant. So I was thinking that since any point in $\mathbb{C}$ can ...
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$B=\{t\in[-T,T]:|f(\sigma+it)|>\ell\}$, does it hold that $|B|>\epsilon 2T$

Fix a $T\in\mathbb{R}$ and a $\sigma\in\mathbb(0,\infty)$ consider the set $$B=\{t\in[-T,T]:|f(\sigma+it)|>\ell\}$$ where $$f:\{s\in\mathbb{C} : \Re (s)>0\}\to\mathbb{C}$$ is a uniformly bounded ...
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367 views

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 $\{n\sqrt{2}\}$ ...
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184 views

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 ...
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1answer
82 views

If almost-periodic function is not identically zero, then it is not in L2

I have an $\mathbb{R}_+ \to \mathbb{R}$ function $f(t)$, which is a combination of sums and products of $\sin$ and $\cos$ functions of incommensurable frequencies. Thus $f(t)$ is a quasiperiodic ...
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97 views

Almost periodic functions

Given a locally compact Abelian group $G$, consider the Banach algebra $l^{\infty}(G)$. Is there some condition such that the algebra $AP(G) \subset l^{\infty}(G)$ of all almost-periodic functions is ...
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1answer
141 views

Fourier transform of an aperiodic function

Given a square-integrable function $f(t)\in \mathbb{R}$ defined on $t \in \mathbb R$ which is both Aperiodic in the sense that $f(t)=f(t+T)$ if and only if $T=0$ Discretely valued in the sense that $...
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1answer
79 views

Regularity of map on the torus

Let $\mathbb{T}^n=\mathbb{R}^n / \mathbb{Z}^n$ and $u:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^1$ function such that there exist $\omega_1, \dots, \omega_n \in \mathbb{Z}$ ($\mathbb{Z}$-linearly ...
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63 views

Periodicity of given modular equation

I want to find the period of repetition of the following sequence $$\left( \frac{x^2}{100}*\left (\frac{2x}{10} \,mod\,\,1\right) \right)\,mod \,1$$ My answer and actual answers are not matching. ...
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30 views

Continuous extension of spectrum 'borders' of the Schroedinger operator

Let $L = -\Delta + u$ operate on $C^\infty_c(\mathbb{R})\subset L^2$ and define its domain by taking the selfadjoint extension of $C^\infty_c(\mathbb{R})$, for $u:\mathbb{R}\to\mathbb{R}$ a quasi ...
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1answer
139 views

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-...