# Fourier-Stieltjes series

From Encyclopedia of Mathematics

A series

where for

(the integrals are taken in the sense of Stieltjes). Here is a function of bounded variation on . Alternatively one could write

(*) |

If is absolutely continuous on , then (*) is the Fourier series of the function . In complex form the series (*) is

where

Moreover,

and will be bounded. If , then is continuous on . There is a continuous function for which does not tend to as . The series (*) is summable to by the Cesàro method , , almost-everywhere on .

#### 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=16120

This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article