# Questions tagged [periodic-functions]

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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 ?
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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 $f$: f^\star(\alpha,\beta ) = \sum_{...
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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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### 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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### 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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### 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 ...
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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 $$P(f '(x)) = Q(f(x))$$ ...
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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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### 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 (...
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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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### 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 ...
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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 ...
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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 ...
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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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### 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 ...
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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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### Evaluating an integral of a periodic function. It's positive?

My purpose is to show that this integral  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,...
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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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### 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 ...
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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 ...
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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 ...
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