Bochner–Riesz mean

The Bochner–Riesz mean is a summability method often used in harmonic analysis when considering convergence of Fourier series and Fourier integrals. It was introduced by Salomon Bochner as a modification of the Riesz mean.

Define

Let ${\displaystyle f}$ be a periodic function, thought of as being on the n-torus, ${\displaystyle \mathbb {T} ^{n}}$, and having Fourier coefficients ${\displaystyle {\hat {f}}(k)}$ for ${\displaystyle k\in \mathbb {Z} ^{n}}$. Then the Bochner–Riesz means of complex order ${\displaystyle \delta }$, ${\displaystyle B_{R}^{\delta }f}$ of (where ${\displaystyle R>0}$ and ${\displaystyle {\mbox{Re}}(\delta )>0}$) are defined as

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