= Summability kernel on the circle
{title2=$\int k_n\,dm=1,\quad\sup_n\|k_n\|_1<\infty,\quad\int_{|t|\geq\delta}|k_n|\,dm\to0$}
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 $2\|f\|_B$ and the vanishing tails control its complement. Thus $\|f*k_n-f\|_B\to0$. The nonnegative <Fejér kernel> is an example with $L^1$ norm one.
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