Welcome to MathOverflow

Question

0

Let $f(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Let $\mu$ be a compactly supported Borel measure (not necessarily positive) on $\mathbb{R}^n$. Define $$ \tilde{\mu}(\xi) = \int e^{i (\xi_1 f(x_1) + \ldots + \xi_n f(x_n))} \mu(dx). $$ I have estimates of the form $|\tilde{\mu}(\xi)| \leq C(\xi)$ for any $\xi \in \mathbb{C}^n$. My question is if it possible to find the similar estimates on Fourier transform $$ \hat{\mu}(\xi) = \int e^{i \xi x} \mu(dx) $$ using given estimates on $\tilde{\mu}(\xi)$?

edited Oct 18 at 10:13

asked Oct 18 at 9:31

Nimza
472●1●7

	Do you want $f$ to be $\mathbb{R}^n\to\mathbb{R}$ or $\mathbb{R}\to\mathbb{R}$? Under the first, your first displayed equation has a type error. – Willie Wong Oct 18 at 10:04
	Yes, thank you, it was an errata. – Nimza Oct 18 at 10:19

Willie Wong · Answer 1 · 2012-10-18 10:03:20Z UTC

1

As stated, no. Let $f \equiv 0$, and $\mu$ with $\int \mu = 0$. You then have $|\tilde{\mu}(\xi)| = 0$ for any $\xi$, and you have no control over the non-zero frequencies $\hat{\mu}$.

answered Oct 18 at 10:03

Willie Wong
6,987●20●43

Of course it is'n true in general. There may be some conditions on $f$ and $\mu$ which garantee the possibility of estimation of Fourier transform. I don't know about such conditions and I would like to know. For example, for function $f(x) = x^{3}$. – Nimza Oct 18 at 10:19

Alexandre Eremenko · Answer 2 · 2012-10-18 19:14:46Z UTC

1

If your $f$ is one-to-one, make the change of the variable in your integral $y_i=f(x_i)$. You will obtain a new measure $\mu_1$, also with compact support, and your $\tilde\mu$ will be the ordinary Fourier transform of this new measure $\mu_1$.

answered Oct 18 at 19:14

Alexandre Eremenko
10.6k●1●24●39

Yes, the task is exactly to find estimates on $\mu$ Fourier transform if we have estimates on $\mu_1 = \phi_{*}\mu$ Fourier transform, where $phi(x_1,\ldots,x_n) = (f(x_1),\ldots,f(x_n))$ and $\phi_{*}\mu$ is a pushforward of measure $\mu$. – Nimza Oct 18 at 19:34

1

I am not sure what kind of estimates you are talking about. If you mean the rate of decrease at infty, this is related to the smoothness of your measure. Then the answer depends on the smoothness of $f$. can you be more specific? – Alexandre Eremenko Oct 18 at 23:10

I'm talking about similar estimates of Fourier transform $\hat{\mu}(\xi)$. I want to find such function $K(\xi)$ that $|\hat{\mu}(\xi)| \leq K(\xi)$. I don't want any estimates on infinity so methods like stationary phase e.t.c. can't help. – Nimza Oct 20 at 9:10

Estimates for Fourier transform

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.