0
votes
0answers
138 views

Establishing an upper bound for a dyadic average of a function in $ {L^{p}}([0,1)) $

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1)) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j} \stackrel{\text{df}}{=} \left[ ...
2
votes
1answer
134 views

To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$

I wants to understand the integrals of the form $$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$ where $\mu(\alpha)$ is a non decreasing function ...
1
vote
2answers
107 views

Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and ...
1
vote
0answers
122 views

Are the two functions equal? If not, what relation are they?

We have $$A=\frac{1}{2\pi}\int_{-\infty}^{\infty}\prod_{i=1}^{\infty}\frac{\sin(v2^{-k})}{v2^{-k}}e^{ivx}dx$$ $$B=\frac{1}{2}\sum_{k=1}^{n}\cos (\pi kx)\prod_{i=1}^{m}\frac{\sin(\pi k2^{-i})}{\pi ...
0
votes
0answers
288 views

Help with an irregular integral

I am looking for help with doing the following integral : $$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in ...
0
votes
1answer
281 views

Fourier coefficients of a rational function

Any ideas how to compute or to approximate integral $$\int_{0}^{1}\frac{(x+a)^{2q}(x+b)^{2q}}{(x-1)^{4q}+x^{4q}}\exp({-2\pi i x y})dx$$ where $q \in \mathbb{N}$ and $a,b =-2,-1,0,1$, $y \in (0,1)$
0
votes
1answer
405 views

($n$-dimensional) Inverse Fourier transform of $\frac{1}{\| \mathbf{\omega} \|^{2\alpha}}$

Note: I first posted question on math.stackexchange and I got one reply, which was a bit helpful (I'm still trying to understand it fully), but did not explore the two solution cases that I mentioned. ...
3
votes
1answer
2k views

Fourier transforms and the Kurzweil-Henstock integral

Is it possible to define Fourier transforms on locally compact commutative groups using the Kurzweil-Henstock integral instead of the Lebesgue integral?