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.

Definition

Define

(ξ)+={ξ,if ξ>00,otherwise.

Let f be a periodic function, thought of as being on the n-torus, 𝕋n, and having Fourier coefficients f^(k) for kn. Then the Bochner–Riesz means of complex order δ, BRδf of (where R>0 and Re(δ)>0) are defined as

BRδf(θ)=kn|k|R(1|k|2R2)+δf^(k)e2πikθ.

Analogously, for a function f on n with Fourier transform f^(ξ), the Bochner–Riesz means of complex order δ, SRδf (where R>0 and Re(δ)>0) are defined as

SRδf(x)=|ξ|R(1|ξ|2R2)+δf^(ξ)e2πixξdξ.

Application to convolution operators

For δ>0 and n=1, SRδ and BRδ 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 Lp spaces is much simpler than the problem of "regular" almost everywhere convergence of Fourier series/integrals (corresponding to δ=0).

In higher dimensions, the convolution kernels become "worse behaved": specifically, for

δn12

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 δ and which p the Bochner–Riesz means of an Lp function converge in norm. This issue is of fundamental importance for n2, since regular spherical norm convergence (again corresponding to δ=0) fails in Lp when p2. This was shown in a paper of 1971 by Charles Fefferman.[1]

By a transference result, the n and 𝕋n problems are equivalent to one another, and as such, by an argument using the uniform boundedness principle, for any particular p(1,), Lp norm convergence follows in both cases for exactly those δ where (1|ξ|2)+δ is the symbol of an Lp bounded Fourier multiplier operator.

For n=2, that question has been completely resolved, but for n3, it has only been partially answered. The case of n=1 is not interesting here as convergence follows for p(1,) in the most difficult δ=0 case as a consequence of the Lp boundedness of the Hilbert transform and an argument of Marcel Riesz.

Define δ(p), the "critical index", as

max(n|1/p1/2|1/2,0).

Then the Bochner–Riesz conjecture states that

δ>δ(p)

is the necessary and sufficient condition for a Lp bounded Fourier multiplier operator. It is known that the condition is necessary.[2]

References

  1. Fefferman, Charles (1971). "The multiplier problem for the ball". Annals of Mathematics 94 (2): 330–336. doi:10.2307/1970864. 
  2. Ciatti, Paolo (2008) (in en). Topics in Mathematical Analysis. World Scientific. p. 347. ISBN 9789812811066. https://books.google.com/books?id=u9glY7i6R2UC&pg=PA347. 

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.