Questions tagged [periodic-functions]
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37
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31
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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
1
vote
1
answer
90
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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 ...
- 410
2
votes
1
answer
159
views
The sum of $q^{-2}$ over nonzero Gaussian integers
I'm reading about the Weierstrass zeta function. In this context,
$\phi(z)=\zeta(z)-\pi\bar{z}$
is periodic over the lattice
$$\mathcal{L}=\{a+bi\mid a,b\in\mathbb{Z}\}.$$
If we take $w\in\mathcal{L}\...
- 31
4
votes
1
answer
450
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Are the irrotational and solenoidal parts of a smooth vector field linearly independent?
Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. It is well known using Helmholtz decomposition that we can decompose $\textbf{F}$ into two vector fields in $V$: $$\textbf{F} = ...
- 185
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3
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219
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Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?
Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let
$$
F_a(t) = \sum_{k \in \mathbb Z} f(t+ak)
$$
be the ...
- 105
0
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0
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155
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Solving a nonlinear differential equation
I need to solve the following equation:
$$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)}$$
with $\eta>1$.
Undoubtedly, the differential ...
- 43
2
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0
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55
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Name of functions that are the discrete sum of a periodic discrete function
Research I am currently engaged in involves the usage of discrete functions of the following form.
Let $a$ be a finite sequence of integers with length $n$ with $a^\infty = a \circ a \circ \dots$ ...
- 21
3
votes
1
answer
300
views
Fourier series of $e^{(\cos(\pi x) - m)^2}$
I'm looking for the Fourier coefficient of a "periodic Gaussian", which writes
$$
f(x) = e^{-\frac{1}{2s}(\cos(\pi x) - m)^2}
$$
It is a real even 2-periodic function, so its Fourier ...
- 31
13
votes
1
answer
606
views
Would efficient factoring have any *other* useful applications?
This question is certainly somewhat opinion-based, but hopefully not hopelessly so.
The granddaddy of all applications for an efficient period finding or factoring capability (e.g. Shor's algorithm) ...
- 1,223
-1
votes
1
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76
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Fundamental of a signal
Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$
or any reasonable Euclidean norm such that bounded ...
- 1,852
2
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0
answers
70
views
Periodic functions on groups
I'll give you some context: We can say that a (classic) periodic function $f:\mathbb{R}\rightarrow\mathbb{R}$ with period $\omega$ is an invariant function under the cyclic subgroup $G=\langle\omega\...
- 171
3
votes
0
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147
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Negative eigenvalue for a periodic Sturm-Liouville problem
Let $f \in C^{\infty}([0, 2\pi])$ be a smooth function and consider the following periodic Sturm-Liouville problem:
$$\begin{cases} u''(x) + f(x)u(x) = - \lambda u(x) \\
u(0) = u(2\pi) \\
u'(0) = u'(2\...
- 2,023
3
votes
1
answer
427
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What is the big-O time complexity of computing $1/N$ to $\log_{2}(N)$ bits of precision?
I am considering large integer values of $N$ (100 or more digits in base-$10$).
In my algorithm, I need to be able to compute the reciprocal of $N$ with enough precision that the repetend will have ...
- 133
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0
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105
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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 ...
- 3,968
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2
answers
468
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When is the periodisation of a function continuous?
Consider a function $f\in\mathcal{C}_0(\mathbb{R})$, where $\mathcal{C}_0(\mathbb{R})$ denotes the space of bounded continuous functions vanishing at infinity. I am interested in the $T$-periodisation ...
- 31
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votes
0
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130
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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( ...
- 2,142
2
votes
1
answer
113
views
$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$ have periodic solution $\iff\; \frac 1T \int_0 ^ T f (t) dt \in g (\mathbb R) $
I have a research work concerning the equation: $$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$$
f and g are defined and continuous in $\mathbb R$ and with values in $\mathbb R$.
...
- 1,503
3
votes
1
answer
695
views
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
3
votes
0
answers
118
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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
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161
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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 ...
- 703
1
vote
0
answers
46
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 $...
3
votes
1
answer
238
views
Smoothing a periodic function of two variables
Let $F \colon \mathbb{R}^2 \to \mathbb{R}^n$ be a $C^{1}$-function 1-periodic in each variable, so it can be considered as a function on the flat torus $\mathbb{T}^2 = \mathbb{R}^2 / \mathbb{Z}^2$. We ...
- 748
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votes
1
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534
views
Periodic function $f$ for which $f(x^2)$ is periodic too
There is the following question which was asked multiple times on Math.SE (e.g. here and here) without any final result:
Question: Is there a periodic function $f:\Bbb R \to\Bbb R$ of smallest ...
- 11.7k
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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
4
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0
answers
379
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
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0
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24
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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
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2
answers
708
views
A conjecture about $\lfloor n!\cdot q/e\rfloor-\,!n\cdot q$
I was thinking about this question asked at Math.SE, when I came up with the following conjecture.
For every $q\in\mathbb Q$ consider a sequence $s_n^{(q)}$ (terms within the sequence are indexed by $...
- 6,599
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0
answers
189
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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
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votes
1
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707
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Does there exist a nonconstant, periodic, real analytic function with period 1 and rational Maclaurin coefficients?
Does there exist a nonconstant, real analytic function $f \colon \mathbb{R} \to \mathbb{R}$ such that $f$ is periodic with period 1 and whose Maclaurin coefficients are all rational?
(The function $\...
- 870
4
votes
1
answer
183
views
Periodic functions over different lattices in $\mathbb R^d$ are linearly independent [closed]
I have the following claim that I think have been proved by someone, but I can not find the reference, hence I would like to ask for help. Here is the claim:
Let $f_1, \ldots, f_n$ be continuous ...
- 320
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votes
1
answer
746
views
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 ...
- 899
4
votes
1
answer
225
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Under what conditions can interval exchanges be approximated by periodic maps?
Under what conditions can an interval exchange be approximated by periodic maps? (in the weak topology for the Lebesgue measure on $[0,1]$ ).
Are there non-trivial examples of periodically ...
- 393
1
vote
1
answer
235
views
Convergence of a sum to the integral
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a 1-periodic function. I am looking about the conditions on $(a,b)\in\mathbb{R}^2$ such that we have the property :
$$\frac{1}{n}\sum_{\ell=0}^{n-1}f(a+b\...
- 65
1
vote
1
answer
599
views
Discretizing a cosine function?
I'd like to start by noting that for some fixed natural $N$ basis functions for my system will be generated by $f(k,x)$ as defined and explained here or in numerous other sources:
$$f(k,x) = \sqrt2 \...
- 221
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2
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1k
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Is there a periodic function without minimum period such that all the possible periods are irrationals? [closed]
Let $f:\mathbb R\to\mathbb R$ be a periodic function. We say $f$ is without minimum period if, $\forall t$ such that $f(x+t)=f(x)\forall x$, there is a $t'$ such that $0<t'<t$ and $f(x+t')=f(x)\...
- 178
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votes
1
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266
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Least common period of a finite sum of exponentials
Hello,
I have come across the function
$f(t) = \sum_{j=1}^n c_j e^{2 \pi i a_j t}$
with $c_j \in \mathbb{C}$, $c_j\neq 0$ and $a_j\in\mathbb{R}$, $a_j \neq 0$ for $j=1,...,n$, and the $a_j$ ...
- 349
14
votes
5
answers
2k
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What is $\sum (x+\mathbb{Z})^{-2}$?
This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges ...
- 12.5k