Difference between revisions of "Fejér summation method"
Ulf Rehmann (talk | contribs) m (moved Fejér summation method to Fejer summation method: ascii title) |
Ulf Rehmann (talk | contribs) m (moved Fejer summation method to Fejér summation method over redirect: accented title) |
(No difference)
|
Revision as of 07:54, 26 March 2012
A summation method of arithmetical averages (cf. Arithmetical averages, summation method of), applied to the summation of Fourier series. It was first applied by L. Fejér [1].
The Fourier series
![]() | (1) |
of a function is summable by the Fejér summation method to a function
if
![]() |
where
![]() | (2) |
and the are the partial sums of (1).
If is a point of continuity of
or a discontinuity of the first kind, then its Fourier series at that point is Fejér summable to
or to
, respectively. If
is continuous on some interval
, then its Fourier series is uniformly Fejér summable on every segment
; and if
is continuous everywhere, then the series is summable to
uniformly on
(Fejér's theorem).
This result was strengthened by H. Lebesgue [2], who proved that for every summable function , its Fourier series is almost-everywhere Fejér summable to
.
The function
![]() |
![]() |
is called the Fejér kernel. It can be used to express the Fejér means (2) of in the form
![]() |
References
[1] | L. Fejér, "Untersuchungen über Fouriersche Reihen" Math. Ann. , 58 (1903) pp. 51–69 |
[2] | H. Lebesgue, "Recherches sur la convergence de séries de Fourier" Math. Ann. , 61 (1905) pp. 251–280 |
[3] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |
[4] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |
Comments
See also Cesàro summation methods.
Fejér summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fej%C3%A9r_summation_method&oldid=22413