Fejér sum

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

$$ \sigma _ {n} ( f, x) = \ { \frac{1}{n + 1 } } \sum _ {k = 0 } ^ { n } s _ {k} ( f, x) = $$

$$ = \ { \frac{a _ {0} }{2} } + \sum _ {k = 1 } ^ { n } \left ( 1 - { \frac{k}{n + 1 } } \right ) ( a _ {k} \cos kx + b _ {k} \sin kx), $$

where $ a _ {k} $ and $ b _ {k} $ are the Fourier coefficients of the function $ f $.

If $ f $ is continuous, then $ \sigma _ {n} ( f, x) $ converges uniformly to $ f ( x) $; $ \sigma _ {n} ( f, x) $ converges to $ f ( x) $ in the metric of $ L $.

If $ f $ belongs to the class of functions that satisfy a Lipschitz condition of order $ \alpha < 1 $, then

$$ \| f ( x) - \sigma _ {n} ( f, x) \| _ {c} = \ O \left ( { \frac{1}{n ^ \alpha } } \right ) , $$

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

$$ \| f ( x) - \sigma _ {n} ( f, x) \| _ {c} = \ o \left ( { \frac{1}{n} } \right ) $$

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.