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
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