Fejér kernel

In mathematics, the Fejér kernel is used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity.

The Fejér kernel is defined as

:$F\_n(x)\; =\; frac\{1\}\{n\}\; sum\_\{k=0\}^\{n-1\}D\_k(x),$

where $D\_k(x)$ is the "k"th order Dirichlet kernel. It can also be written in a closed form as

:$F\_n(x)\; =\; frac\{1\}\{n\}\; left(frac\{sin\; frac\{n\; x\}\{2\{sin\; frac\{x\}\{2\; ight)^2$,

where this expression is defined. It is named after the Hungarian mathematician Lipót Fejér (1880–1959).

The important property of the Fejér kernel is $F\_n(x)\; ge\; 0$. The convolution "F_n" is positive: for $f\; ge\; 0$ of period $2\; pi$ it satisfies

:$0\; le\; (f*F\_n)(x)=frac\{1\}\{2pi\}int\_\{-pi\}^pi\; f(y)\; F\_n(x-y),dy,$

and, by the Hölder's inequality, $|F\_n*f\; |\_\{L^p(\; [-pi,\; pi]\; )\}\; le\; |f|\_\{L^p(\; [-pi,\; pi]\; )\}$ for every $0\; le\; p\; le\; infty$ or continuous function $f$;moreover, $f*F\_n\; ightarrow\; f$ for every $f\; in\; L^p(\; [-pi,\; pi]\; )$ () or continuous function $f$.

ee also

* Fejér's theorem
* Gibbs phenomenon
* Charles Jean de la Vallée-Poussin