# 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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### 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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### 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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### “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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### 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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### 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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### 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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### 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 ...
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 ...