# Fejér summation method

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)