Questions tagged [periodic-functions]
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17
questions with no upvoted or accepted answers
5
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135
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The space of periodizable tempered distribution
The periodization operator $\mathrm{Per}$ is defined for a Schwartz function $\varphi \in \mathcal{S}(\mathbb{R})$ as
\begin{equation}
\mathrm{Per} \{ \varphi \} (x) = \sum_{n \in \mathbb{Z}} \varphi( ...
- 2,174
4
votes
0
answers
381
views
Evaluating an integral of a periodic function. It's positive?
My purpose is to show that this integral
\begin{equation}
I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du\,\,,\,\,x,...
- 141
3
votes
0
answers
158
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Negative eigenvalue for a periodic Sturm-Liouville problem
Let $f \in C^{\infty}([0, 2\pi])$ be a smooth function and consider the following periodic Sturm-Liouville problem:
$$\begin{cases} u''(x) + f(x)u(x) = - \lambda u(x) \\
u(0) = u(2\pi) \\
u'(0) = u'(2\...
- 2,085
3
votes
0
answers
122
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An elliptic function built from a log-theta-function integral?
I'm studying the apparent ellipticity of $$\Theta(z,a):=\frac{\exp(\tfrac2a\int_0^a\ln\vartheta(x+z)\,dx)}{\vartheta(z)\vartheta(z+a)},\tag1$$with $a$ is a free parameter and \begin{align}\vartheta(z)&...
- 149
3
votes
0
answers
105
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Probability of Intersection of Randomly Shifted Pulses
Defining a function $f_{a,T}:\mathbb R \to \{0,1\}$ to be $T$-periodic ($\forall x: f_{a,T}(x)=f_{a,T}(x+T)$), with $a\in[0,T]$ such that $\forall x\in [0,T] : f_{a,T}(x)= 1 \iff x\in [0,a]$.
Given ...
- 255
2
votes
0
answers
56
views
Name of functions that are the discrete sum of a periodic discrete function
Research I am currently engaged in involves the usage of discrete functions of the following form.
Let $a$ be a finite sequence of integers with length $n$ with $a^\infty = a \circ a \circ \dots$ ...
- 21
2
votes
0
answers
73
views
Periodic functions on groups
I'll give you some context: We can say that a (classic) periodic function $f:\mathbb{R}\rightarrow\mathbb{R}$ with period $\omega$ is an invariant function under the cyclic subgroup $G=\langle\omega\...
- 171
2
votes
0
answers
111
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Can all (inverse) trigonometric functions with periodic iterates be characterized?
I wonder whether all (composites of) trigonometric and inverse trigonometric functions with periodic functional iterations can be found. In order to specify what I mean by that, let's introduce some ...
- 4,575
1
vote
0
answers
46
views
Can we bound the squared Gaussian curvature of genus three triply periodic minimal surfaces?
Assume that $\mathcal{M}$ is a balanced triply periodic minimal surface of genus 3, embedded in a flat torus $T^3=\mathbb{R}^3/\Lambda$ for a lattice $\Lambda$ with volume 1. I want to understand the ...
1
vote
0
answers
52
views
Mean of a periodic velocity field and trajectory displacement bound
Suppose $u(t,x)$ is a smooth velocity field on $[0,\infty)\times \mathbb{R}$ and periodic in space, i.e., $u(t,0)=u(t,1)$ $\forall t$. Assume that $\int_0^1 u(t,x) \,dx = c$, independent of time. Let $...
1
vote
0
answers
196
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Quantification of the extent of periodicity in a time series using fractal analyses
I need metrics to quantify and compare the extent of periodicity between any two given time series, considering the time series were "almost periodic". By "almost periodic" I mean: if I were to take ...
- 111
0
votes
0
answers
42
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Equating two Fourier Series with different periods
For some $\tau\in(0,1)$, let $f : (-\infty,0]\times [0,\tau] \rightarrow \mathbb{C}$ and $g:[0,\infty)\times[0,1]\rightarrow\mathbb{C}$ with $$f(0,t)=g(0,t)\text{ for }t\in [0,\tau]$$ be two smooth ...
- 149
0
votes
0
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61
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Construct a continuous function $f(x)$ periodic with period $2\pi$ such that the Fourier series of $f(x)$ is divergent at $x = 0$
The following question was asked on Math Stack Exchange by me 15 days ago. I used a bounty, but still no response.So I am posting the question here
Here is the link of the question here
problem ...
- 109
0
votes
0
answers
38
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Lower bound of infinite sum of shifts
This is an extension of https://mathoverflow.net/posts/452526.
So, it appears that $\sin(xt)$ serves as a better lower bound instead of a linear equation, in fact the series appears to uniformly ...
- 410
0
votes
0
answers
156
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Solving a nonlinear differential equation
I need to solve the following equation:
$$y'(t)+2[\cos y(t)+\Omega(t)]=0,$$
where
$$\Omega(t)=-2\eta +\frac{2(\eta^2-1)}{\eta-\cos(4\sqrt{\eta^2-1}t)}$$
with $\eta>1$.
Undoubtedly, the differential ...
- 43
0
votes
0
answers
164
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What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?
I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by
a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking,
in that the residuals from the ...
- 723
0
votes
0
answers
27
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Difference between $W^{k,p}([0,1]^d)$ and $W^{k,p}(\mathbb{T}^d)$
Let $\mathbb{T}^d \sim \mathbb{R}^d/\mathbb{Z}^d$. I know that $$W^{k,2}(\mathbb{T})\equiv\{f\in W^{k,2}([0,1]); f^{(i)}(0) = f^{(i)}(1), i = 0, \ldots, k-1\}.$$
Are there similar characterizations ...
- 176