# Fourier-Stieltjes series

A series

$${ \frac{a _ {0} }{2} } + \sum _ {n = 1 } ^ \infty ( a _ {n} \cos nx + b _ {n} \sin nx),$$

where for $n = 0, 1 \dots$

$$a _ {n} = \ { \frac{1} \pi } \int\limits _ { 0 } ^ { {2 } \pi } \cos nx dF ( x),\ \ b _ {n} = \ { \frac{1} \pi } \int\limits _ { 0 } ^ { {2 } \pi } \sin nx dF ( x)$$

(the integrals are taken in the sense of Stieltjes). Here $F$ is a function of bounded variation on $[ 0, 2 \pi ]$. Alternatively one could write

$$\tag{* } dF ( x) \sim \ { \frac{a _ {0} }{2} } + \sum _ {n = 1 } ^ \infty ( a _ {n} \cos nx + b _ {n} \sin nx).$$

If $F$ is absolutely continuous on $[ 0, 2 \pi ]$, then (*) is the Fourier series of the function $F ^ { \prime }$. In complex form the series (*) is

$$dF ( x) \sim \ \sum _ {n = - \infty } ^ { {+ } \infty } c _ {n} e ^ {inx} ,$$

where

$$c _ {n} = \ { \frac{1}{2 \pi } } \int\limits _ { 0 } ^ { {2 } \pi } e ^ {-} inx dF ( x).$$

Moreover,

$$F ( x) - c _ {0} x \sim \ C _ {0} + \sum _ { \begin{array}{c} n = - \infty \\ n \neq 0 \end{array} } ^ \infty \frac{c _ {n} }{in } e ^ {inx} ,$$

and $\{ c _ {n} \}$ will be bounded. If $c _ {n} \rightarrow 0$, then $F$ is continuous on $[ 0, 2 \pi ]$. There is a continuous function $F$ for which $c _ {n}$ does not tend to $0$ as $n \rightarrow + \infty$. The series (*) is summable to $F ^ { \prime } ( x)$ by the Cesàro method $( C, r)$, $r > 0$, almost-everywhere on $[ 0, 2 \pi ]$.

#### References

 [1] A. Zygmund, "Trigonometric series" , 1 , Cambridge Univ. Press (1988)
How to Cite This Entry:
Fourier-Stieltjes series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier-Stieltjes_series&oldid=46961
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article