# 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.

## Definition

Define

$\displaystyle{ (\xi)_+ = \begin{cases} \xi, & \mbox{if } \xi \gt 0 \\ 0, & \mbox{otherwise}. \end{cases} }$

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 $\displaystyle{ R \gt 0 }$ and $\displaystyle{ \mbox{Re}(\delta) \gt 0 }$) are defined as

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

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

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

## Application to convolution operators

For $\displaystyle{ \delta \gt 0 }$ and $\displaystyle{ n=1 }$, $\displaystyle{ S_R^\delta }$ and $\displaystyle{ B_R^\delta }$ 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 $\displaystyle{ L^p }$ spaces is much simpler than the problem of "regular" almost everywhere convergence of Fourier series/integrals (corresponding to $\displaystyle{ \delta = 0 }$).

In higher dimensions, the convolution kernels become "worse behaved": specifically, for

$\displaystyle{ \delta \leq \tfrac{n-1}{2} }$

the kernel is no longer integrable. Here, establishing almost everywhere convergence becomes correspondingly more difficult.

## Bochner–Riesz conjecture

Another question is that of for which $\displaystyle{ \delta }$ and which $\displaystyle{ p }$ the Bochner–Riesz means of an $\displaystyle{ L^p }$ function converge in norm. This issue is of fundamental importance for $\displaystyle{ n \geq 2 }$, since regular spherical norm convergence (again corresponding to $\displaystyle{ \delta = 0 }$) fails in $\displaystyle{ L^p }$ when $\displaystyle{ p \neq 2 }$. This was shown in a paper of 1971 by Charles Fefferman.[1]

By a transference result, the $\displaystyle{ \mathbb{R}^n }$ and $\displaystyle{ \mathbb{T}^n }$ problems are equivalent to one another, and as such, by an argument using the uniform boundedness principle, for any particular $\displaystyle{ p \in (1, \infty) }$, $\displaystyle{ L^p }$ norm convergence follows in both cases for exactly those $\displaystyle{ \delta }$ where $\displaystyle{ (1-|\xi|^2)^{\delta}_+ }$ is the symbol of an $\displaystyle{ L^p }$ bounded Fourier multiplier operator.

For $\displaystyle{ n=2 }$, that question has been completely resolved, but for $\displaystyle{ n \geq 3 }$, it has only been partially answered. The case of $\displaystyle{ n=1 }$ is not interesting here as convergence follows for $\displaystyle{ p \in (1, \infty) }$ in the most difficult $\displaystyle{ \delta = 0 }$ case as a consequence of the $\displaystyle{ L^p }$ boundedness of the Hilbert transform and an argument of Marcel Riesz.

Define $\displaystyle{ \delta (p) }$, the "critical index", as

$\displaystyle{ \max( n|1/p - 1/2| - 1/2, 0) }$.

Then the Bochner–Riesz conjecture states that

$\displaystyle{ \delta \gt \delta (p) }$

is the necessary and sufficient condition for a $\displaystyle{ L^p }$ bounded Fourier multiplier operator. It is known that the condition is necessary.[2]

## References

1. Fefferman, Charles (1971). "The multiplier problem for the ball". Annals of Mathematics 94 (2): 330–336. doi:10.2307/1970864.
2. Ciatti, Paolo (2008) (in en). Topics in Mathematical Analysis. World Scientific. p. 347. ISBN 9789812811066.