# 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)