New answers tagged fourier-transform
1
vote
Does this distribution exist?
Denote $f(t) = \frac{\sin t}{t}$. Then your identity is that for any $u,v\in\mathbb{R}^2$,
$$
\frac{\hat W(u) \hat W(v)}{\hat W(u+v)} = f(u\times v)
$$
Therefore, for any $u,v,w\in\mathbb{R}^2$,
$$
\...
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5
votes
What is the intuition behind applying the Mellin Transform to prime distribution?
The prime numbers encode multiplicative structure. Similarly, the von Mangoldt function $\Lambda$ encodes additive structure, as expressed by the identity
$\log n = \sum_{d \mid n} \Lambda(d).$
...
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7
votes
Accepted
What is the intuition behind applying the Mellin Transform to prime distribution?
You observed that the Mellin transform is used to diagonalize dilations.
One can indeed motivate its presence in the theory of prime numbers in this form. However, we're not interested in ...
- 137k
7
votes
What is the intuition behind applying the Mellin Transform to prime distribution?
This is too thin an answer really, but it's too long for a comment - I expect there'll be much better answers soon enough but hopefully this is at least a start:
If you have any sum, you wonder what ...
- 1,166
3
votes
Does this distribution exist?
It seems your equation implies $\hat{W}(0,\mu_1)\hat{W}(0,\mu_2)=\hat{W}(0,\mu_1+\mu_2)$, which would mean that $\hat{W}(0,\mu)= e^{c\mu}$. This is not the Fourier transform of a valid marginal ...
- 178k
7
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Is $\frac{|t|}{e^{a|t|}-e^{-b|t|}}$ the Fourier transform of a positive function
No for $0<a<b$ your function has a global maximum at $\pm x_0 \neq 0$. Then, if it was $\hat{\varphi} = f $ for some positive function $\varphi$, $f(x-y)$ would be a positive semidefinite kernel....
1
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Is $\frac{|t|}{e^{a|t|}-e^{-b|t|}}$ the Fourier transform of a positive function
Comment
The inverse Fourier transform of $\phi_{1,2}$ looks like this
Is there any reason to think there exist $a,b$ where it has different behavior?
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1
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Examining the Hilbert transform of functions over the positive real line
The answer to the second question is negative as well. Take for example $g$ supported in $(-\infty,-1)$ and discontinuous in some point. If $f$ is supported in $\mathbb{R}_+$ and $y,z<-1$ it holds ...
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