Questions tagged [almost-periodic-function]
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31 questions
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Which ζ-function controls the density of base=2 Miller–Rabin pseudoprimes?
Background: the density d (≈ inverse distance between elements of the set) of primes is “controlled by zeros of ζ-function” in the sense that $(d(x) - T(x))·\log(x)/√x$ is an almost periodic function ...
- 455
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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
$$
\frac{C_1}{t^2} \leq \min_{|x|\leq t} V(x)\leq \frac{C_2}{t^2}
$$
...
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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 ...
- 402
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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 ...
- 593
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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:
$$
f(z)=\frac{\sum_i a_i e^{i(q_{i,x}x+q_{i,y}y)}}{\sum_i ...
- 219
5
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Intuition for almost periodic solution and Poincaré recurrence theorem
I would like to ask a question that I had asked yesterday on the site math.stackexchange and I still have not received an answer.
Suppose that we have a PDE that admit a solution $u$ that can be ...
- 93
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3
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Can we calculate the probability that $f(x)$ is positive for a random $x\in(0,m)$ as $m\to\infty$? (uniform distribution)
Following my previous question here, I have this function
$$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\...
user215601
5
votes
1
answer
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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 ...
- 536
1
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1
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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, \...
- 1,322
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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)?
...
- 19.7k
3
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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 ...
- 147
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1
answer
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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 \...
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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 ...
- 785
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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 ...
- 173
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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) ...
- 185
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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 ...
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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 ...
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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 ...
- 161
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1
answer
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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 $f$ ...
- 1,045
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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}}\,...
- 663
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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 ...
- 798
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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 ...
- 1,045
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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 ...
- 798
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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 ...
- 785
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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 ...
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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 ...
- 4,422
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1
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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 \...
- 2,560
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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 ...
- 1,010
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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 ...
- 1,084
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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{...
- 2,600
4
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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) \: ...
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