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 $\tilde{\mu}(\xi)$?