10
votes
Accepted
Can we characterize a periodic function by the compactness of the set of its translates?
I would also like to focus on the case $\nu =1$ (to avoid some slightly annoying but probably trivial bookkeeping issues).
Suppose that $\{f_t\}$ is compact. This shows first of all that $f$ is ...
- 24.2k
6
votes
What are the almost periodic functions on the complex plane?
Yes to the first question: the almost periodic functions on any LCA group are the uniform limits of linear combinations of characters. In the case of $\mathbb{R}^2$ these are the functions $e^{i(ax + ...
- 42.8k
6
votes
Accepted
Does such a function exist?
Define $f(t,x\mathbf v_1+y\mathbf v_2)=e^{2\pi i(x/g(t)+yg(t))}$.
Then $f(t,\mathbf x+j\mathbf v_1)=e^{2\pi ij/g(t)}f(t,\mathbf x)$ and similarly $f(t,\mathbf y+k\mathbf v_2)=e^{2\pi ikg(t)}f(t,\...
- 23.2k
5
votes
Accepted
Bohr compactification as a topological compactification
As Francois Ziegler answered, for a locally compact group $G$, the Bohr compactification of $G$ is a compactification in the usual sense iff $G$ is compact. This is true with no further restrictions.
...
- 11.6k
5
votes
Bohr compactification as a topological compactification
The answer is no: unless $G$ is compact, its subspace topology inside $bG$ (called its Bohr topology) is much weaker than its own. So $\iota:G\to bG$, while continuous with dense image, is not an ...
- 31.5k
5
votes
Accepted
Intuition for almost periodic solution and Poincaré recurrence theorem
The definition which you wrote can be translated in words like this.
You evaluate the solution at time $t$. Then you fix a small distance $\epsilon>0$. Then you can find a time length $\ell(\...
- 4,459
4
votes
Accepted
About understanding manifold structure on WAP compactification of $\Bbb{C} \rtimes \Bbb{T}$
Let me recall that the WAP compactification of a locally compact group is a semi-topological compact monoid ("semi-topological" means that both left and right multiplications are continuous, but maybe ...
- 11.6k
4
votes
Almost periodicity of Bessel functions
Maple gave me this...
$$
J_0(x) = \left( {\frac {\sin \left( x \right) }{\sqrt {\pi}}}+{\frac {\cos
\left( x \right) }{\sqrt {\pi}}} \right) x^{-1/2}+ \left( -{\frac {\cos \left( x \right) }{8\sqrt {\...
- 41.1k
4
votes
Distribution of distances of successive zeros of $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$
The peaks arise from envelopes that arise when the function value nearly repeats. The peaks are at the roots of the implicit equation $\pm {\rm cos}(x)\pm {\rm cos}(\alpha x)\pm {\rm cos}(\beta x)=0$ ...
4
votes
What does the unique mean on weakly almost periodic functions look like?
This is not a full answer but merely an additional perspective that was too long for a comment.
Instead of the general WAP functions, let us discuss the smaller class of matrix coefficient in a ...
- 426
3
votes
Distribution of distances of successive zeros of $f(x)={\rm cos}(x)+{\rm cos}(\alpha x)+{\rm cos}(\beta x)$
Here is how I think of this problem from a ergodic theory point of view. Because the system is ergodic for long time the point $(t [2\pi] , \alpha t [2\pi], \beta t [2\pi] )$ should span $[2\pi]^3 $ ...
- 4,361
3
votes
Accepted
Is $f(n)=\cos(2\pi\theta(1+2+\ldots+n))$ almost periodic?
I believe the answer is no, for the same sort of reason that $g(t)=e^{it^2}$ is not almost periodic on $\mathbf R$.
(Indeed, the latter would mean that given $\varepsilon>0$ there is a $T$ such ...
- 31.5k
3
votes
Accepted
Approximation of quasi-periodic function by trigonometric polynomials
Yes. This is Satz XIV, p. 160 of Harald Bohr, Zur Theorie der Fastperiodischen Funktionen: II, Acta Math. 46 (1925) 101-214. (Page 162 attributes the quasi-periodic case to Bohl.)
- 31.5k
3
votes
Accepted
Long time average of solution to ODE with almost periodic structure
For the funciton you gave, the limit is 0, but my proof below only gives convergence as $1/\log s$. If $f=2-\sin(2\pi x)-\sin(2\pi Lx)$ where $L$ is Louiville's constant (or some appropriately chosen ...
- 1,160
2
votes
Accepted
On $B^1$ and $B^2$ almost-periodic functions
The answer depends on what exactly you mean by the question. The subtle thing about $B^p$ is that it represents not functions by classes, and the classes depend on $p$.
This is a problem since if you ...
- 2,071
2
votes
Accepted
A limit related to quasi-periodic function
This cannot work, and in fact the Diophantine properties of $\sqrt{2}$ play no role here. Basically, the reason is that the condition imposes a kind of behavior on $(x,\sqrt{2}x)$ that is much too ...
- 24.2k
1
vote
Accepted
Bounds of periodic functions formed from infinite series of shifts
This is a little too long for a comment, but I hope it might help.
Note that the function $$f_a(t,x) = \sum_{m=-\infty}^\infty (-1)^m k_a(t+mx)$$ $$k_a(t) = (t+a t^3) e^{-t^2} $$ for $a \geq 0$ ...
- 524
1
vote
Can we calculate the probability that $f(x)$ is positive for a random $x\in(0,m)$ as $m\to\infty$? (uniform distribution)
While Mathematica's command NIntegrate[] will likely produce an output with a few correct digits, it will not guarantee any of them.
To get such a guarantee, you can partition the square $[0,2\pi]\...
- 128k
1
vote
Can we calculate the probability that $f(x)$ is positive for a random $x\in(0,m)$ as $m\to\infty$? (uniform distribution)
A sample Mathematica code to find the area of the region given in Anthony Quas's comment is:
...
- 17.2k
1
vote
Accepted
Distribution of an almost periodic trigonometric polynomial
Map the torus $\mathbb T^2$ to $2 \mathbb D$ by the projection $\pi: (\theta, \phi) \mapsto e^{i\theta} + e^{i\phi}$. The trajectory is uniformly distributed on $\mathbb T^2$, and what you have is ...
- 54.2k
1
vote
Accepted
Dominated convergence for quasi-periodic functions
The proposition is indeed true. To see this one uses an alternative characterization of quasi-periodic functions. Let $\omega_1, \dots, \omega_m\in \mathbb{R}$ a $\mathbb{Z}$-basis of the frequency ...
- 1,045
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