# Questions tagged [periodic-functions]

The periodic-functions tag has no usage guidance.

48
questions

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### 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
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1
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132
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### 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_{...

2
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0
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38
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### 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 ...

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0
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### Fourier series of function with zero of infinite order?

It is easy to give examples of positive periodic functions with zeros of order $n$ where $n$ is a natural number and whose Fourier series expansion is explicit (powers of cosine will do). I wonder if ...

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47
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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 ...

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0
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171
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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 ...

4
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1
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158
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### 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
$$P(f '(x)) = Q(f(x))$$
...

8
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1
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172
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### 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 ...

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1
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151
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### 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
(...

5
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1
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216
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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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0
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47
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### 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 ...

0
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0
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40
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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 ...

1
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1
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107
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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 ...

2
votes

1
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183
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### 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}\...

4
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1
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511
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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} = ...

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3
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252
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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 ...

0
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0
answers

162
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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 ...

2
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0
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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 $a^\infty = a \circ a \circ \dots$ ...

3
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1
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391
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### 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 ...

13
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1
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652
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### 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) ...

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1
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77
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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 ...

2
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0
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74
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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\...

3
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0
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159
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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
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1
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473
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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 ...

2
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0
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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
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2
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589
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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 ...

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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
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1
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### $x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$ 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 $\mathbb R$.
...

3
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1
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814
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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 ...

3
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0
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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)&...

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165
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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 ...

1
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0
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### 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 $...

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1
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262
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### 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 ...

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1
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635
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### 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 ...

3
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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 ...

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0
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381
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### 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,...

0
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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 ...

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2
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### 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 $...

1
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0
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197
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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 ...

8
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1
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727
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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 $\...

4
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1
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195
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### 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 ...

5
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1
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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 ...

4
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1
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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 ...

1
vote

1
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239
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### 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\...

1
vote

1
answer

624
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### 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 \...

1
vote

2
answers

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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)\...

0
votes

1
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271
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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 $a_j$ ...

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votes

5
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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 ...