Questions tagged [periodic-functions]

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Lower bound of infinite sum of shifts

This is an extension of So, it appears that $\sin(xt)$ serves as a better lower bound instead of a linear equation, in fact the series appears to uniformly ...
Bobby Ocean's user avatar
1 vote
1 answer

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 ...
Bobby Ocean's user avatar
2 votes
1 answer

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}\...
isz's user avatar
  • 31
4 votes
1 answer

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} = ...
MrPie 's user avatar
  • 185
6 votes
3 answers

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 ...
user975628's user avatar
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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 ...
Young Q's user avatar
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2 votes
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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 $a^\infty = a \circ a \circ \dots$ ...
Jondow8's user avatar
  • 21
3 votes
1 answer

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 ...
gaspardb's user avatar
13 votes
1 answer

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) ...
tparker's user avatar
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-1 votes
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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 ...
Arthur B's user avatar
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2 votes
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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\...
PQH's user avatar
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3 votes
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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\...
Eduardo Longa's user avatar
3 votes
1 answer

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 ...
mlchristians's user avatar
2 votes
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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 ...
Max Muller's user avatar
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3 votes
2 answers

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 ...
MatthieuMeo's user avatar
5 votes
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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( ...
Goulifet's user avatar
  • 2,142
2 votes
1 answer

$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$ 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 $\mathbb R$. ...
Pascal's user avatar
  • 1,503
3 votes
1 answer

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 ...
DrManhattan's user avatar
3 votes
0 answers

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)&...
Antonio DJC's user avatar
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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 ...
Paul B. Slater's user avatar
1 vote
0 answers

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 $...
Stephen C. Preston's user avatar
3 votes
1 answer

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 ...
demolishka's user avatar
11 votes
1 answer

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 ...
M. Winter's user avatar
  • 11.7k
3 votes
0 answers

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 ...
Dudi Frid's user avatar
  • 255
4 votes
0 answers

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,...
ppooppii's user avatar
  • 141
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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 ...
Housen's user avatar
  • 176
15 votes
2 answers

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 $...
Vladimir Reshetnikov's user avatar
1 vote
0 answers

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 ...
np20's user avatar
  • 111
8 votes
1 answer

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 $\...
Ken Fan's user avatar
  • 870
4 votes
1 answer

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 ...
sweehong's user avatar
  • 320
5 votes
1 answer

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 ...
Greg Zitelli's user avatar
4 votes
1 answer

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 ...
Mostafa's user avatar
  • 393
1 vote
1 answer

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\...
Christophe's user avatar
1 vote
1 answer

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 \...
Francis's user avatar
  • 221
1 vote
2 answers

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)\...
alberto.bosia's user avatar
0 votes
1 answer

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 $a_j$ ...
Eric Foxall's user avatar
14 votes
5 answers

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 ...
Greg Muller's user avatar
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