Fourier-Stieltjes series

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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