# Tagged Questions

**3**

votes

**1**answer

229 views

### Can the Fourier series of a continuous function diverge on an uncountable set of measure zero?

I know that there exist continuous function $f: [0,2\pi]\rightarrow\mathbb{R}$ whose Fourier series diverges at all rational points of $[0,2\pi]$(c.f. Katznelson).We also know that the set of ...

**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}}.
...

**1**

vote

**2**answers

316 views

### Asymptotics of Fourier coefficients of power-type functions

I would like to understand the asymptotic behaviour of the Fourier coefficients of
power type functions
$f(t) = |t|^{-\alpha} 1_{[-\pi, \pi]} \qquad 0 < \alpha<1.$
I suppose this is a classic ...

**2**

votes

**0**answers

475 views

### What square-summable sequences are “sinc-summable”?

$\operatorname{sinc} : \mathbb{R} \to \mathbb{R} \;\;$ is defined by $\;\; \operatorname{sinc}(x) \; = \; \begin{cases} 1 & \text{if }\:\;x=0 \\ \\ \frac{\operatorname{sin}(x)}x & \text{else} ...

**28**

votes

**3**answers

3k views

### Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

Hi.
Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway...
Consider this problem. I've been trying to find a formula to expand the "regular iteration" ...

**0**

votes

**1**answer

508 views

### Series of squared Fourier coefficients

Hi, if the Fourier series development of $g(t)$ (periodic, $C^\infty$) is
$$
g(t)=\sum_{-\infty}^{+\infty}a_n e^{in\omega t}
$$
does the series
$$
\sum_{-\infty}^{+\infty}\frac{a_n^2}{n^2}?
$$
...

**3**

votes

**1**answer

897 views

### Characterizations of a linear subspace associated with Fourier series

Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace
a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable ...