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

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

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

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

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

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

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.

References

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

Further reading

• 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.