# 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 .

How to Cite This Entry:
Fejér sum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fej%C3%A9r_sum&oldid=46912
This article was adapted from an original article by S.A. Telyakovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article