Questions tagged [periodic-functions]
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49 questions
2 votes
0 answers
112 views
Is this Knapsack looking recurrence always periodic?
Let $(a_n)_{n\geq 1}$ be a sequence of integers such that for some constant $N$ and all $n>N$ we have $$ a_n = -\max_{i+j=n}(a_i + a_j). $$ Is it true that any such sequence is eventually periodic? ...
- 163
4 votes
0 answers
159 views
Periodic Distributions and Banach Completions of $\mathcal{S}(\mathbb{R})$
Let $N$ be a norm on the Schwartz space $\mathcal{S}(\mathbb{R})$, and let $B(\mathbb{R})$ be the Banach space completion of $\mathcal{S}(\mathbb{R})$ under $N$, with the assumption that $B(\mathbb{R})...
- 2,602
11 votes
2 answers
565 views
Preperiod of powers of matrices modulo m
Let $A$ be a square matrix with integer entries and let $m$ be a positive integer. From the pigeonhole principle it follows easily that the sequence $$I,A, A^2, A^3,\; \dots \pmod m$$ is eventually ...
- 113
0 votes
0 answers
67 views
Existence of periodic solutions of the Hill's Equation
Let $\mathbb{K}:=\mathbb{R}$ or $\mathbb{C}$. Let $T>0$ and let $p\in C(\mathbb{R};\mathbb{K})$ be $T$-periodic. Let us consider the Hill's Equation $$ (H)\ \ \ \ \ \ \ x''(t)+p(t)x(t)=0. $$ I am ...
- 985
2 votes
1 answer
246 views
What is the fastest algorithm for classical period finding?
Let $N$ be a positive integer, and choose an integer $a$ such that $\gcd(a,N)=1$. Then $a^r \equiv 1 \,\text{mod}\, N$ for some $r$. What is the current fastest classical algorithm for finding the ...
- 591
1 vote
1 answer
215 views
Does any such family of functions exist?
Is there a sequence of non-zero bounded smooth functions $f_1,f_2,\ldots,f_k$ so that $$\sum_{I=1}^k \cos(f_i)= \cos\left(\sum_{i=1}^k f_i \right).$$ And what about the infinite case ?
-1 votes
1 answer
202 views
Does transforming a periodic function imply periodicity
Let $f(x,y)$ be a periodic function for every fixed $y = \beta$ with respect to $x$ in the domain $x\in \mathbb{R}$ and consider this transform of $f$: \begin{equation} f^\star(\alpha,\beta ) = \sum_{...
3 votes
0 answers
91 views
What circumstances guarantee a p-adic affine conjugacy map will be a rational function?
Let $\Bbb Q_p$ be a p-adic field and let any element $x$ of $\Bbb Q_p$ be associated with a unique element of $\Bbb Z_p$ via the quotient / equivalence relation $\forall n\in\Bbb Z:p^nx\sim x$ Then in ...
- 406
5 votes
1 answer
266 views
When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?
$\newcommand{\cl}{\operatorname{cl}}\newcommand{\sl}{\operatorname{sl}}\newcommand{\cm}{\operatorname{cm}}\newcommand{\sm}{\operatorname{cm}}$Consider the differential equation $$P(f '(x)) = Q(f(x))$$ ...
- 799
8 votes
1 answer
415 views
Approximation of triply periodic minimal surfaces with trigonometric level sets
Some triply periodic minimal surfaces are known to be approximated by trigonometric level sets very accurately. To see this, let's sample a gyroid scaled to the bounding box $[0, 1]^3$ exactly through ...
- 451
1 vote
1 answer
200 views
To find a $2\pi$-periodic function with a property
I recently came across the following question in my research, and I don't know how to proceed this problem. Question: How to find a function $g(x)$ such that it satisfies (1) $2\pi$ periodic (2) odd (...
- 405
5 votes
1 answer
272 views
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 $$ \frac{C_1}{t^2} \leq \min_{|x|\leq t} V(x)\leq \frac{C_2}{t^2} $$ ...
user124759
2 votes
0 answers
83 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
1 answer
168 views
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
2 votes
1 answer
273 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
549 views
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 $V$: $$\textbf{F} = ...
user353746
6 votes
3 answers
344 views
Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?
Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let $$ F_a(t) = \sum_{k \in \mathbb Z} f(t+ak) $$ be the ...
- 137
0 votes
0 answers
183 views
Solving a nonlinear differential equation
I need to solve the following equation: $$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)}$$ with $\eta>1$. Undoubtedly, the differential ...
- 43
2 votes
0 answers
72 views
Name of functions that are the discrete sum of a periodic discrete function
Research I am currently engaged in involves the usage of discrete functions of the following form. Let $a$ be a finite sequence of integers with length $n$ with $a^\infty = a \circ a \circ \dots$ ...
- 21
3 votes
1 answer
502 views
Fourier series of $e^{(\cos(\pi x) - m)^2}$
I'm looking for the Fourier coefficient of a "periodic Gaussian", which writes $$ f(x) = e^{-\frac{1}{2s}(\cos(\pi x) - m)^2} $$ It is a real even 2-periodic function, so its Fourier ...
- 31
13 votes
1 answer
760 views
Would efficient factoring have any *other* useful applications?
This question is certainly somewhat opinion-based, but hopefully not hopelessly so. The granddaddy of all applications for an efficient period finding or factoring capability (e.g. Shor's algorithm) ...
- 1,321
-1 votes
1 answer
97 views
Fundamental of a signal
Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$ or any reasonable Euclidean norm such that bounded ...
- 1,912
2 votes
0 answers
103 views
Periodic functions on groups
I'll give you some context: We can say that a (classic) periodic function $f:\mathbb{R}\rightarrow\mathbb{R}$ with period $\omega$ is an invariant function under the cyclic subgroup $G=\langle\omega\...
- 171
3 votes
0 answers
195 views
Negative eigenvalue for a periodic Sturm-Liouville problem
Let $f \in C^{\infty}([0, 2\pi])$ be a smooth function and consider the following periodic Sturm-Liouville problem: $$\begin{cases} u''(x) + f(x)u(x) = - \lambda u(x) \\ u(0) = u(2\pi) \\ u'(0) = u'(2\...
- 2,127
2 votes
1 answer
562 views
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 ...
- 99
2 votes
0 answers
137 views
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 ...
- 5,025
4 votes
2 answers
868 views
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 ...
- 41
5 votes
0 answers
196 views
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( ...
- 2,602
2 votes
1 answer
145 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
976 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
189 views
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
0 votes
0 answers
201 views
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 ...
- 733
1 vote
0 answers
69 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
328 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 ...
- 902
11 votes
1 answer
756 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 ...
- 14.5k
3 votes
0 answers
121 views
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 ...
- 265
4 votes
0 answers
398 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
0 votes
0 answers
56 views
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
13 votes
2 answers
783 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 $...
- 6,063
1 vote
0 answers
248 views
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
8 votes
1 answer
828 views
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
240 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
5 votes
1 answer
963 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 ...
- 1,194
4 votes
1 answer
264 views
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 ...
- 403
1 vote
1 answer
284 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 : $$\frac{1}{n}\sum_{\ell=0}^{n-1}f(a+b\...
- 65
1 vote
1 answer
674 views
Discretizing a cosine function?
I'd like to start by noting that for some fixed natural $N$ basis functions for my system will be generated by $f(k,x)$ as defined and explained here or in numerous other sources: $$f(k,x) = \sqrt2 \...
- 261
1 vote
2 answers
1k views
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)\...
- 188
0 votes
1 answer
300 views
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 views
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 ...
- 13.2k