Bochner–Riesz mean

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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 integrals. It was introduced by Salomon Bochner as a modification of the Riesz mean.


(\xi)_+ = \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)_+^\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)_+^\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}_+ 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.


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

Further reading[edit]

  • Lu, Shanzhen (2013). Bochner-Riesz Means on Euclidean Spaces (First ed.). World Scientific. ISBN 978-981-4458-76-4. 
  • Grafakos, Loukas (2008). Classical Fourier Analysis (Second ed.). Berlin: Springer. ISBN 978-0-387-09431-1. 
  • Grafakos, Loukas (2009). Modern Fourier Analysis (Second ed.). Berlin: Springer. ISBN 978-0-387-09433-5. 
  • Stein, Elias M. & Murphy, Timothy S. (1993). Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton University Press. ISBN 0-691-03216-5.