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

$(\xi)_%2B = \begin{cases} \xi, & \mbox{if } \xi > 0 \\ 0, & \mbox{otherwise}. \end{cases}$

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

$B_R^\delta f(\theta) = \underset{|k| \leq R}{\sum_{k \in \mathbb{Z}^n}} \left( 1- \frac{|k|^2}{R^2} \right)_%2B^\delta \hat{f}(k) e^{2 \pi i k \cdot \theta}.$

Analogously, for a function $f$ on $\mathbb{R}^n$ with Fourier transform $\hat{f}(\xi)$ , the Bochner–Riesz means of complex order $\delta$ , $S_R^\delta f$ (where $R > 0$ and $\mbox{Re}(\delta) > 0$ ) are defined as

$S_R^\delta f(x) = \int_{|\xi| \leq R} \left(1 - \frac{|\xi|^2}{R^2} \right)_%2B^\delta \hat{f}(\xi) e^{2 \pi i x \cdot \xi}\,d\xi.$

For $\delta > 0$ and $n=1$ , $S_R^\delta$ and $B_R^\delta$ may be written as convolution operators, where the convolution kernel is an approximate identity. As such, in these cases, considering the almost everywhere convergence of Bochner–Riesz means for functions in $L^p$ spaces is much simpler than the problem of "regular" almost everywhere convergence of Fourier series/integrals (corresponding to $\delta = 0$ ). In higher dimensions, the convolution kernels become more "badly behaved" (specifically, for $\delta \leq \tfrac{n-1}{2}$ , the kernel is no longer integrable) and establishing almost everywhere convergence becomes correspondingly more difficult.

Another question is that of for which $\delta$ and which $p$ the Bochner–Riesz means of an $L^p$ function converge in norm. This is of fundamental importance for $n \geq 2$ , since regular spherical norm convergence (again corresponding to $\delta = 0$ ) fails in $L^p$ when $p \neq 2$ . This was shown in a paper of 1971 by Charles Fefferman.^[1] By a transference result, the $\mathbb{R}^n$ and $\mathbb{T}^n$ problems are equivalent to one another, and as such, by an argument using the uniform boundedness principle, for any particular $p \in (1, \infty)$ , $L^p$ norm convergence follows in both cases for exactly those $\delta$ where $(1-|\xi|^2)^{\delta}_%2B$ is the symbol of an $L^p$ bounded Fourier multiplier operator. For $n=2$ , this question has been completely resolved, but for $n \geq 3$ , it has only been partially answered. The case of $n=1$ is not interesting here as convergence follows for $p \in (1, \infty)$ in the most difficult $\delta = 0$ case as a consequence of the $L^p$ boundedness of the Hilbert transform and an argument of Marcel Riesz.

References

^ Fefferman, Charles (1971). "The multiplier problem for the ball". Annals of Mathematics 94 (2): 330–336. doi:10.2307/1970864.

Bochner–Riesz mean

References

Further reading