I understand the definition of characteristic functions used in
probability theory:
For a random Variable $X$ with probability density function $f_X$ the characteristic function is defined as:
$$\phi_X(t) = E(\exp(itX)) = \int_{\mathbf{R}} e^{\mathrm{i}tx}f_X(x)\, dx.$$
I read that
[…] the characteristic function in nonprobabilistic contexts is called the
fourier transform
(Page 342, of Probability and Measure. P.Billingsley 3rd editon).
But I still can't see this from their definitions, because the fourier transform $f^*$ of a function $f$ is defined as follows:
$$ f^*(t) =
\int_{\mathbf{R}^n} f(x)\,e^{-\mathrm{i} t \cdot x} \,\mathrm{d} x$$
If the function $f$ is a density, for e.g. $f=f_X$, then the fourier transform can be written as:
$$ f^*_X = \int_{\mathbf{R}^n} f_X(x)\,e^{-\mathrm{i} t \cdot x} \,\mathrm{d} x \neq \int_{\mathbf{R}} e^{\mathrm{i}tx}f_X(x)\, dx = \phi_X(t) $$
The definitions differ with a minus, why can they than be used both for the same thing (e.g. for deconvolution)?
Best Answer
Actually, the Fourier transform can be defined in both ways by using $e^{-\mathrm{i} t \cdot x}$ or $e^{\mathrm{i} t \cdot x}$. They are essentially the same, just like you can call either $i$ or $-i$ the imaginary unit.
By the same taken, you can define the characteristic functions via the Fourier transform or the inverse Fourier transform depending on your choice.