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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)?

Ferdi
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Giuseppe
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    Yes, as you note $f_X^* = \phi_X(-t)$. They can each be used for the things you use the other for, *as long as you keep the change of sign straight* (and in some definitions, different constants out the front) when going the other way (inverting the transformation). – Glen_b Dec 02 '12 at 23:32
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    There are two square roots of $-1$. Mathematicians use one of them and call it $i$; engineers use the other and call it $j$. Since $j = -i$, the characteristic function $\int e^{itx} f(x) dx$ used by mathematicians is exactly the same as the Fourier transform $\int e^{-jtx} f(x) dx$ used by engineers – Dilip Sarwate Dec 03 '12 at 02:17
  • related: http://math.stackexchange.com/questions/58163/dft-why-are-the-definitions-for-inverse-and-forward-commonly-switched/60423#60423 – leonbloy Jan 02 '13 at 02:20

1 Answers1

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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.

chaohuang
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