# Fejér summation method

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

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 .

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 , 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 How to Cite This Entry:
Fejér summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fej%C3%A9r_summation_method&oldid=18036
This article was adapted from an original article by I.I. Volkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article