= Powered-cosine approximate identity on a torus
{title2=$K_N(s,t)=c_N(2+\cos s+\cos t)^N$}
On the <torus> $\mathbb T^2$ with normalized <Haar measure>, choose $c_N$ so that $\int K_N=1$. The kernel is a nonnegative <trigonometric polynomial>. Its base $2+\cos s+\cos t$ has unique maximum four at the origin. Outside any fixed neighborhood its maximum is $q<4$, while in a smaller neighborhood of positive measure it is at least some $q'>q$. Normalization consequently bounds the exterior kernel by $C(q/q')^N$, which tends to zero. <Convolution> with these kernels gives uniform <trigonometric polynomial> approximation to every <continuous function> on the torus.
Back to article page
