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**5**

votes

**0**answers

103 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 ...

**4**

votes

**1**answer

116 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 ...

**4**

votes

**1**answer

228 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 ...

**4**

votes

**1**answer

186 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 ...

**1**

vote

**1**answer

189 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 :
...

**1**

vote

**1**answer

185 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 ...

**1**

vote

**2**answers

576 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 ...

**0**

votes

**1**answer

211 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 $a_j$ ...

**8**

votes

**4**answers

1k 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 ...