Summability kernel

In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability kernels are related to approximation of the identity; definitions of an approximation of identity vary, but sometimes the definition of an approximation of the identity is taken to be the same as for a summability kernel.

Let ${\displaystyle \mathbb {T} :=\mathbb {R} /\mathbb {Z} }$. A summability kernel is a sequence ${\displaystyle (k_{n})}$ in ${\displaystyle L^{1}(\mathbb {T} )}$ that satisfies

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