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

$$\frac12+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)\label{1}\tag{1}$$

of a function $f\in L(-\pi,\pi)$ is summable by the Fejér summation method to a function $s$ if

$$\lim_{n\to\infty}\sigma_n(x)=s(x),$$

where

$$\sigma_n(x)=\frac{1}{n+1}\sum_{k=0}^ns_k(x),\label{2}\tag{2}$$

and the $s_k(x)$ are the partial sums of \eqref{1}.

If $x$ is a point of continuity of $f$ or a discontinuity of the first kind, then its Fourier series at that point is Fejér summable to $f(x)$ or to $(f(x+0)+f(x-0))/2$, respectively. If $f$ is continuous on some interval $(a,b)$, then its Fourier series is uniformly Fejér summable on every segment $[\alpha,\beta]\subset(a,b)$; and if $f$ is continuous everywhere, then the series is summable to $f$ uniformly on $[-\pi,\pi]$ (Fejér's theorem).

This result was strengthened by H. Lebesgue [2], who proved that for every summable function $f$, its Fourier series is almost-everywhere Fejér summable to $f$.

The function

$$K_n(x)=\frac{1}{n+1}\sum_{k=0}^n\left(\frac12+\sum_{\nu=1}^k\cos\nu x\right)\equiv$$

$$\equiv\frac{1}{2(n+1)}\left(\frac{\sin(n+1)x/2}{\sin x/2}\right)^2$$

is called the Fejér kernel. It can be used to express the Fejér means \eqref{2} of $f$ in the form

$$\sigma_n(x)=\frac1\pi\int_{-\pi}^\pi f(x+u)K_n(u)du.$$

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.