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The '''Bochner–Riesz mean''' is a [[summability method]] often used in [[harmonic analysis]] when considering convergence of [[Fourier series]] and [[Fourier inversion theorem|Fourier integrals]]. It was introduced by [[Salomon Bochner]] as a modification of the [[Riesz mean]].
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Define
 
:<math>(\xi)_+ = \begin{cases} \xi, & \mbox{if }  \xi > 0  \\ 0, & \mbox{otherwise}. \end{cases}</math>
 
Let <math>f</math> be a periodic function, thought of as being on the [[n-torus]], <math>\mathbb{T}^n</math>, and having Fourier coefficients <math>\hat{f}(k)</math> for <math>k \in \mathbb{Z}^n</math>.  Then the Bochner–Riesz means of complex order <math>\delta</math>, <math>B_R^\delta f</math> of (where <math>R > 0</math> and <math>\mbox{Re}(\delta) > 0</math>) are defined as
 
:<math>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}.</math>
 
Analogously, for a function <math>f</math> on <math>\mathbb{R}^n</math> with Fourier transform <math>\hat{f}(\xi)</math>, the Bochner–Riesz means of complex order <math>\delta</math>, <math>S_R^\delta f</math> (where <math>R > 0</math> and <math>\mbox{Re}(\delta) > 0</math>) are defined as
 
:<math>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.</math>
 
For <math>\delta > 0</math> and <math>n=1</math>, <math>S_R^\delta</math> and <math>B_R^\delta</math> may be written as [[convolution]] operators, where the convolution kernel is an [[mollifier|approximate identity]]. As such, in these cases, considering the [[almost everywhere convergence]] of Bochner–Riesz means for functions in <math>L^p</math> spaces is much simpler than the problem of "regular" almost everywhere convergence of Fourier series/integrals (corresponding to <math>\delta = 0</math>). In higher dimensions, the convolution kernels become more "badly behaved" (specifically, for <math>\delta \leq \tfrac{n-1}{2}</math>, the kernel is no longer integrable) and establishing almost everywhere convergence becomes correspondingly more difficult.
 
Another question is that of for which <math>\delta</math> and which <math>p</math> the Bochner–Riesz means of an <math>L^p</math> function converge in norm.  This is of fundamental importance for <math>n \geq 2</math>, since regular spherical norm convergence (again corresponding to <math>\delta = 0</math>) fails in <math>L^p</math> when <math>p \neq 2</math>.  This was shown in a paper of 1971 by [[Charles Fefferman]].<ref>{{Cite journal |first=Charles |last=Fefferman |title=The multiplier problem for the ball |journal=[[Annals of Mathematics]] |volume=94 |issue=2 |year=1971 |pages=330–336 |doi=10.2307/1970864 }}</ref> By a transference result, the <math>\mathbb{R}^n</math> and <math>\mathbb{T}^n</math> problems are equivalent to one another, and as such, by an argument using the [[uniform boundedness principle]], for any particular <math>p \in (1, \infty)</math>, <math>L^p</math> norm convergence follows in both cases for exactly those <math>\delta</math> where <math>(1-|\xi|^2)^{\delta}_+</math> is the [[Fourier multiplier|symbol]] of an <math>L^p</math> bounded [[Fourier multiplier]] operator. For <math>n=2</math>, this question has been completely resolved, but for <math>n \geq 3</math>, it has only been partially answered. The case of <math>n=1</math> is not interesting here as convergence follows for <math>p \in (1, \infty)</math> in the most difficult <math>\delta = 0</math> case as a consequence of the <math>L^p</math> boundedness of the [[Hilbert transform]] and an argument of [[Marcel Riesz]].
 
==References==
{{Reflist}}
 
==Further reading==
*{{Cite book |first=Shanzhen |last=Lu |year=2013 |title=Bochner-Riesz Means on Euclidean Spaces |edition=First |publisher=World Scientific |isbn=978-981-4458-76-4 }}
*{{Cite book |first=Loukas |last=Grafakos |year=2008 |title=Classical Fourier Analysis |edition=Second |location=Berlin |publisher=Springer |isbn=978-0-387-09431-1 }}
*{{Cite book |first=Loukas |last=Grafakos |year=2009 |title=Modern Fourier Analysis |edition=Second |location=Berlin |publisher=Springer |isbn=978-0-387-09433-5 }}
*{{Cite book |first=Elias M. |last=Stein |authorlink=Elias M. Stein |lastauthoramp=yes |first2=Timothy S. |last2=Murphy |year=1993 |title=Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals |location=Princeton |publisher=Princeton University Press |isbn=0-691-03216-5 }}
 
{{DEFAULTSORT:Bochner-Riesz Mean}}
[[Category:Means]]
[[Category:Summability methods]]

Latest revision as of 08:03, 4 May 2014

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