Bochner–Riesz mean explained

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

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

Let

f be a periodic function, thought of as being on the n-torus, T ^{n} , and having Fourier coefficients \hat{f}(k) for k \in Z ^{n} . Then the Bochner–Riesz means of complex order \delta , B ^{\delta} _{R} f of (where R > 0 and Re (\delta) > 0 ) are defined asB ^{\delta} _{R} f(\theta) = \underset{|k| \leq R}{\sum _{k \in Z ^{n}} } \left(1- \frac

^2

\right)_+^\delta \hat(k) e^.

Analogously, for a function

f on R ^{n} with Fourier transform \hat{f}(\xi) , the Bochner–Riesz means of complex order \delta , S ^{\delta} _{R} f (where R > 0 and Re (\delta) > 0 ) are defined asS ^{\delta} _{R} f(x) = \int _{|\xi|} \left( 1 - (|\xi| ^{2})/(R ^{2}) \right) ^{\delta} _{+} \hat{f}(\xi) e ^{2} d\xi.

Application to convolution operators

For

\delta > 0 and n =1 , S ^{\delta} _{R} and B ^{\delta} _{R} 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 "worse behaved": specifically, for

\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

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

By a transference result, the

R ^{n} and 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 , inf ty) , 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 , that 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 , inf ty) 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.

Define

\delta (p) , the "critical index", asmax ( n| 1 /p - 1 / 2 | - 1 / 2 , 0 ) .

Then the Bochner–Riesz conjecture states that

\delta > \delta (p)

is the necessary and sufficient condition for a

L ^{p} bounded Fourier multiplier operator. It is known that the condition is necessary.^[2]

Further reading

• Book: Lu, Shanzhen . 2013 . Bochner-Riesz Means on Euclidean Spaces . First . World Scientific . 978-981-4458-76-4 .
• Book: Grafakos, Loukas . 2008 . Classical Fourier Analysis . Second . Berlin . Springer . 978-0-387-09431-1 .
• Book: Grafakos, Loukas . 2009 . Modern Fourier Analysis . Second . Berlin . Springer . 978-0-387-09433-5 .
• Book: Elias M. Stein

. Elias M. . Stein . Elias M. Stein . amp . Timothy S. . Murphy . 1993 . Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals . Princeton . Princeton University Press . 0-691-03216-5 .

Notes and References

1. Charles . Fefferman . The multiplier problem for the ball . . 94 . 2 . 1971 . 330–336 . 10.2307/1970864 . 1970864 .
2. Book: Ciatti . Paolo . Topics in Mathematical Analysis . 2008 . World Scientific . 9789812811066 . 347 . en.