Summability kernel on the circle ($\int k_{n}dm=1,{\sup }_n\Vert k_n{\Vert }_1<\infty ,{\int }_{|t|\ge \delta }|k_n|dm\to 0$)

A sequence of continuous periodic functions satisfying the displayed conditions, with normalized Haar measure $m$ and every $\delta >0$, is a summability kernel. For $f$ in a homogeneous Banach space on the circle, subtract $f\int k_{n}dm$ from its convolution with $k_n$. Translation continuity controls the resulting integral near zero, while boundedness by $2\Vert f{\Vert }_B$ and the vanishing tails control its complement. Thus $\Vert f\ast k_n-f{\Vert }_B\to 0$. The nonnegative Fejér kernel is an example with $L^1$ norm one.