 Powered-cosine approximate identity on a torus

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 $q^{\prime }>q$. Normalization consequently bounds the exterior kernel by $C(q/q^{\prime }{)}^N$, which tends to zero. Convolution with these kernels gives uniform trigonometric polynomial approximation to every continuous function on the torus.