in mathematics, the fejÃ©r kernel is a summability kernel used to express the effect of cesÃ ro summation on fourier series. it is a non-negative kernel, giving rise to an approximate identity. it is named after the hungarian mathematician lipÃ³t fejÃ©r (1880&ndash;1959). thumb|400px|plot of several fejÃ©r kernels ==definition== the fejÃ©r kernel is defined as :f_n(x) = \frac{1}{n} \sum_{k=0}^{n-1}d_k(x), where :d_k(x)=\sum_{s=-k}^k {\rm e}^{isx} is the kth 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}}\right)^2 = \frac{1}{n} \left(\frac{1 - \cos(nx)}{1 - \cos x}\right) , where this expression is defined. the fejÃ©r kernel can also be expressed as :f_n(x)=\sum_{|j|\le n}\left(1-\frac{|j|}{n}\right)e^{ijx}. ==properties== the fejÃ©r kernel is a positive summability kernel. an important property of the fejÃ©r kernel is f_n(x) \ge 0 with average value of 1 . ===convolution=== the convolution fn is positive: for f \ge 0 of period 2 \pi it satisfies :0 \le (f*f_n)(x)=\frac{1}{2\pi}\int_{-\pi}^\pi f(y) f_n(x-y)\,dy. since f*d_n=s_n(f)=\sum_{|j|\le n}\widehat{f}_je^{ijx}, we have f*f_n=\frac{1}{n}\sum_{k=0}^{n-1}s_n(f), which is cesÃ ro summation of fourier series. by young's inequality, :\|f_n*f \|_{l^p([-\pi, \pi])} \le \|f\|_{l^p([-\pi, \pi])} for every 1 \le p \le \infty for f\in l^p. additionally, if f\in l^1([-\pi,\pi]), then :f*f_n \rightarrow f a.e. since [-\pi,\pi] is finite, l^1([-\pi,\pi])\supset l^2([-\pi,\pi])\supset\cdots\supset l^\infty([-\pi,\pi]), so the result holds for other l^p spaces, p\ge1 as well. if f is continuous, then the convergence is uniform, yielding a proof of the weierstrass theorem. * one consequence of the pointwise a.e. convergence is the uniquess of fourier coefficients: if f,g\in l^1 with \hat{f}=\hat{g}, then f=g a.e. this follows from writing f*f_n=\sum_{|j|\le n}\left(1-\frac{|j|}{n}\right)e^{ijt}, which depends only on the fourier coefficients. * a second consequence is that if \lim_{n\to\infty}s_n(f) exists a.e., then \lim_{n\to\infty}s_n(f)=f a.e., since cesÃ ro means f_n*f converge to the original sequence limit if it exists. ==see also== * fejÃ©r's theorem * dirichlet kernel * gibbs phenomenon * charles jean de la vallÃ©e-poussin ==references== 
Fourier_series