Characteristic function and probability density function: Fourier or Inverse Fourier? I have come across two contradicting definitions of characteristics function (CHF). In wikipedia CHF is defined as the inverse Fourier transform (FT) of probability density function (PDF) and at some places (e.g. http://www.math.nus.edu.sg/~matsr/ProbI/Lecture6.pdf) it is defined as FT of the pdf ?
Cordially,
 A: If the density exists, then the characteristic function will be a scaled form of the Fourier transform. There are multiple conventions for the Fourier transform - I use
$$\hat f(t)=\int_{-\infty}^\infty f(x)e^{-2\pi ixt}\,\mathrm d x$$
in which case the characteristic function of a random variable with density $f$ is given by
$$\varphi(t)=E(e^{itX})=\int_{-\infty}^\infty e^{itx}f(x)\,\mathrm dx=\hat f(-t/2\pi)$$
The reason that some people might refer to the inverse Fourier transform is that given suitable regularity assumptions, we have
$$f(x)=\int_{-\infty}^\infty \hat f(t)e^{2\pi i xt}\,\mathrm dt$$
and so if $X$ has density $\hat f$ then its characteristic function is $\varphi(t)=f(t/2\pi)$. This is just a scaled version of the inverse Fourier transform, and the scale in this case is positive. As Did pointed out, taking either of these as definition is unsatisfactory since the characteristic function is defined for any random variable $X$ as $E(e^{itX})$, whereas the Fourier transform definition requires the existence of a density.
A: $$\varphi_X(t)=E(\mathrm e^{\mathrm i\langle t,X\rangle})=\int\mathrm e^{\mathrm i\langle t,x\rangle}\,\mathrm dP_X(x)$$
