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 .
|||L. Fejér, "Untersuchungen über Fouriersche Reihen" Math. Ann. , 58 (1903) pp. 51–69|
|||N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)|
|||A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)|
|||I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian)|
|||V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)|
See also Fejér summation method.
Fejér sum. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fej%C3%A9r_sum&oldid=23270