Fourier-Stieltjes series

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)