Fourier-Stieltjes transform

One of the integral transforms (cf. Integral transform) related to the Fourier transform. Let the function $ F $ have bounded variation on $ (- \infty , + \infty ) $. The function

$$ \tag{* } \phi ( x) = \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \infty } ^ {+\infty } e ^ {-} ixy dF ( y) $$

is called the Fourier–Stieltjes transform of $ F $. The function $ \phi $ determined by the integral (*) is bounded and continuous. Every periodic function $ \phi $ that can be expanded in an absolutely-convergent Fourier series $ \sum _ {- \infty } ^ {+ \infty } a _ {n} e ^ {inx} $ can be written as an integral (*) with $ F ( x) = \sum _ {n \leq x } a _ {n} $.

Formula (*) can be inverted: If $ F $ has bounded variation and if

$$ F ^ \bullet ( x) = \ { \frac{F ( x + 0) + F ( x - 0) }{2} } , $$

then

$$ F ^ \bullet ( x) - F ^ \bullet ( 0) = \ { \frac{1}{\sqrt {2 \pi } } } \int\limits _ {- \infty } ^ {+\infty } \phi ( \xi ) \frac{e ^ {i \xi x } - 1 }{i \xi } \ d \xi ,\ \ x \in (- \infty , + \infty ), $$

where the integral is taken to mean the principal value at $ \infty $.

If one only allows non-decreasing functions of bounded variation as the function $ F $ in formula (*), then the set of continuous functions $ \phi $ thus obtained is completely characterized by the property: For any system of real numbers $ t _ {1} \dots t _ {n} $,

$$ \sum _ {i, j = 1 } ^ { n } \phi ( t _ {i} - t _ {j} ) \xi _ {i} \overline \xi \; _ {j} \geq 0, $$

whatever the complex numbers $ \xi _ {1} \dots \xi _ {n} $( the Bochner–Khinchin theorem). Such functions are called positive definite. The Fourier–Stieltjes transform is extensively applied in probability theory, where the non-decreasing function

$$ P ( x) = \ { \frac{1}{\sqrt {2 \pi } } } F ( x) $$

is subjected to the additional restrictions $ \lim\limits _ {x \rightarrow - \infty } P ( x) = 0 $, $ \lim\limits _ {x \rightarrow + \infty } P ( x) = 1 $ and $ P $ is continuous on the left; it is called a distribution, and

$$ \Phi(x) = \int\limits_{-\infty}^{+\infty} e^{ixy} dP(y) $$

is called the characteristic function (of the distribution $ P $). The Bochner–Khinchin theorem then expresses a necessary and sufficient condition for a continuous function $ \Phi $( for which $ \Phi ( 0) = 1 $) to be the characteristic function of a certain distribution.

The Fourier–Stieltjes transform has also been developed in the $ n $- dimensional case.

References

[1]	S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)
[2]	A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[3]	B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian)