Fourier-Stieltjes transform

One of the integral transforms (cf. Integral transform) related to the Fourier transform. Let the function have bounded variation on . The function

(*)

is called the Fourier–Stieltjes transform of . The function determined by the integral (*) is bounded and continuous. Every periodic function that can be expanded in an absolutely-convergent Fourier series can be written as an integral (*) with .

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

then

where the integral is taken to mean the principal value at .

If one only allows non-decreasing functions of bounded variation as the function in formula (*), then the set of continuous functions thus obtained is completely characterized by the property: For any system of real numbers ,

whatever the complex numbers (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

is subjected to the additional restrictions , and is continuous on the left; it is called a distribution, and

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

The Fourier–Stieltjes transform has also been developed in the -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)