## New answers tagged harmonic-analysis

0
votes

### approximating differentiable functions with double trigonometric polynomials

In my opinion this is a question of MSE level.
For the first question the answer is yes. Because finite trigonometric polynomials is dense in $C^k(\Bbb T^n)$. This is essentially another Weierstrass ...

0
votes

### Fourier series of smooth functions in infinitely many variables

Not sure why this is brought up after some years. But I think you should google "Fourier analysis on infinite torus" or something similar. The first result returns me a paper by Denis Fufaev....

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 $u,v,w\in\mathbb{R}^2$,
$$
\...

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
$\log n = \sum_{d \mid n} \Lambda(d).$
...

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 ...

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 ...

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 ...

4
votes

Accepted

### Radial Fourier transform vs Hankel transform

The function does not have a standard term but is widely understood to be the radial solution of the Helmholtz equation (which is the time-periodic solutions of the wave equation) arrived at by what ...

6
votes

### High dimensional Lusin conjecture

See Michael Lacey's exposition on Carleson's theorem for some discussion of this, particularly Section 9 there.
In higher dimensions one needs to specify the order of summation. If one sums over the ...

0
votes

### Complex sum of squares of vector fields (hypoelliptic operators)

While this question is old, it's perhaps worth noting that something is known: It's known that the situation can be very complicated.
For example, Kohn (Annals of Mathematics, 162 (2005), 943–986) ...

1
vote

### Check an equation on the Heisenberg group $H_1$

First, the reference is: https://www.sciencedirect.com/science/article/pii/0022123682900787
Second, in the notation of the cited paper $H_n={\mathbb C}^n\times {\mathbb R}$. Hence, you must decide if ...

3
votes

Accepted

### On compactly supported functions with prescribed sparse coordinates

This is a rather lazy version of an answer, but I will also indicate how I think a full answer can be produced.
Let $u(x,z)$ be the solution of $-u''+qu=z u$ with $u(0,z)=0$, $u'(0,z)=1$. Then
$$
B_L =...

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