- Fejér kernel
In
mathematics , the Fejér kernel is used to express the effect ofCesàro summation onFourier series . It is a non-negative kernel, giving rise to anapproximate 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 "Fn" 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] ) (1 le p < infty) or continuous function f.ee also
*
Fejér's theorem
*Gibbs phenomenon
*Charles Jean de la Vallée-Poussin
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