Fejér sum

One of the arithmetic means of the partial sums of a Fourier series in the trigonometric system

where and are the Fourier coefficients of the function .

If is continuous, then converges uniformly to ; converges to in the metric of .

If belongs to the class of functions that satisfy a Lipschitz condition of order , then

that is, in this case the Fejér sum approximates at the rate of the best approximating functions of the indicated class. But Fejér sums cannot provide a high rate of approximation: The estimate

is valid only for constant functions.

Fejér sums were introduced by L. Fejér [1].

References

[1]	L. Fejér, "Untersuchungen über Fouriersche Reihen" Math. Ann. , 58 (1903) pp. 51–69
[2]	N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)
[3]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)
[4]	I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)
[5]	V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)

Comments

See also Fejér summation method.