Quantile given Characteristic function

Question

For a random variable defined by the PDF $f(x)$ and CDF $F(x)$,

• Characteristic function $CF(x)$ (if it exists) is given by the Fourier transform of $f(x)$.
• Quantile $Q(x)$ (if it exists) is given by the inverse of $F(x)$.

If both $CF(x)$ and $Q(x)$ exists,

1. What is the quantile given the characteristic function?
2. Is there a relationship between the two similar to the inverse function theorem?

Answer 1

The characteristic function, say $\varphi$, exists for any real-valued variable, even if multivariate, but I'll stick to the univariate case. But it's a function of a dummy variable typically denoted $t$, not $x$. If the PDF exists, it's$$f(x)=\frac{1}{2\pi}\int_{\Bbb R}e^{-itx}\varphi(t)dt.$$More generally, the CDF is$$\frac{1}{2\pi}\int_{\Bbb R}\int_{-\infty}^xe^{-itu}du\varphi(t)dt.$$