# Tagged Questions

**4**

votes

**2**answers

98 views

### Approximate Moment Conditions

It is known in classical probability that if two random variables $X$ and $Y$ obeys
$$\mathbb{E} X^k = \mathbb{E}Y^k, \ \forall \ k \geq 1$$
with additional condition that $\mathbb{E}X^k$ does not ...

**1**

vote

**0**answers

74 views

### The $d$-dimension extension of Bernoulli Polynomial

It is known that Bernoulli polynomial has the following Fourier expansion:
\begin{equation*}
B_{2n}(x) = \frac{(-1)^{n-1}2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{\cos(2k\pi x)}{k^{2n}}.
...

**2**

votes

**1**answer

151 views

### Approximation power of wavelets

The Wikipedia article on Wavelet Transform states that:
Wavelet compression is not good for all kinds of data: transient signal characteristics mean good wavelet compression, while smooth, ...

**2**

votes

**1**answer

204 views

### Bound on trigonometric sum

I want to show that there is some $\gamma(n)=o(n^{-1})$ and some $C(n) \to -\infty$ such that for $\gamma \leq \theta \leq \pi$ we have
$\sum_{k=1}^n -1+\cos(k\theta) \leq C(n)$.
If we rewrite this ...

**5**

votes

**2**answers

889 views

### L1 distance from a trigonometric susbspace

How to check, whether the $L^{1}$ distance between a finite exponential sum $S_{F}(x)=\sum\limits_{n\in F} \exp(inx)$ and the $L^{1}$-closure of subspace $\mathrm{span}\left(\exp(inx): n\in ...

**1**

vote

**0**answers

626 views

### What is the function $\sin(n \omega) / (n \sin \omega)$?

During my work, I encounter the function like $\frac{\sin(n \omega)}{n \sin \omega}$. I'm puzzled and knew nothing about this function before.
Given integer $n>1$, my question is how to find a ...

**6**

votes

**2**answers

835 views

### Non-linear “Fourier analysis”

Call a function of the following form a beep: $e^{-(\frac{x-\alpha}{\beta})^2}\sin(\rho x+\theta)$. Given a real-valued function $f\in L^2(R)$ and a number $n$, I'm interested in the approximating ...

**7**

votes

**2**answers

2k views

### approximate a probability distribution by moment matching

Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...