-1
votes
1answer
92 views

Oddify an even function and vice versa: need a Fourier transform-based formula [closed]

I have constructed an operator that applied to an odd function will give its even counterpart and also an inverse operator that would transform an even function into an odd one. I need a ...
0
votes
0answers
54 views

Where the following interpolation method converges?

In this question about discrete-analytic functions (that is functions, who equal to their Newton series) I asked for a solution for the following problem: Is there a method to extend the notion ...
0
votes
1answer
224 views

Prove that these two definitions of “natural” integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details). The first one is based on ...
3
votes
1answer
245 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
0answers
77 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
2answers
321 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
0answers
497 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
3answers
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
1answer
525 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
1answer
933 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 ...