Bochner–Riesz means

Remarks

Example: for $\delta = 0$, $S^0_R f$ is the spherical partial sum, the inverse Fourier transform of $\hat f$ restricted to the open ball of radius $R$. For $n = 1$ and $\delta = 1$, the multiplier is $(1 - \xi^2/R^2)_+$, which is a continuous function with compact support.

The multiplier of $S^\delta_R$ is the dilate of the multiplier of $S^\delta$ by $R$, and dilations do not change $L^p$ operator norms. So $S^\delta_R$ is bounded on $L^p(\R^n)$ uniformly in $R$ if and only if $(1 - \abs{\xi}^2)_+^\delta$ is an $L^p$ multiplier. In that case, $S^\delta_R f \to f$ in $L^p(\R^n)$ as $R \to \infty$ for every $f \in L^p(\R^n)$, $1 \le p < \infty$.

Non-example of convergence: for $\delta = 0$, $n \ge 2$ and $p \ne 2$, the ball multiplier is not an $L^p$ multiplier, by Fefferman's ball multiplier theorem.

For $\delta > (n - 1)/2$, the kernel of $S^\delta$ is integrable, so $S^\delta$ is bounded on $L^p(\R^n)$ for all $1 \le p \le \infty$. The value $(n - 1)/2$ is the critical index.