Relativistic heat conduction

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

$(\xi {)}_{+}=\{\begin{array}{ll}\xi ,& \text{if }\xi >0\\ 0,& \text{otherwise}.\end{array}$

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 $k\in {\mathbb{\mathbb{Z}}}^n$. Then the Bochner–Riesz means of complex order $\delta$, $B_R^{\delta }f$ of (where $R>0$ and $\text{Re}(\delta )>0$) are defined as

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

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 $\text{Re}(\delta )>0$) are defined as

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

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 $\delta \le \frac{n-1}{2}$, 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 $n\ge 2$, since regular spherical norm convergence (again corresponding to $\delta =0$) fails in $L^p$ when $p\ne 2$. This was shown in a paper of 1971 by Charles Fefferman. 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 $p\in (1,\mathrm{\infty })$, $L^p$ 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 $n\ge 3$, it has only been partially answered. The case of $n=1$ is not interesting here as convergence follows for $p\in (1,\mathrm{\infty })$ 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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Further reading

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534