# 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.

How to Cite This Entry:
Fourier–Stieltjes transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier%E2%80%93Stieltjes_transform&oldid=22450