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

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Define

 

Let   be a periodic function, thought of as being on the n-torus,  , and having Fourier coefficients   for  . Then the Bochner–Riesz means of complex order  ,   of (where   and  ) are defined as

 

Analogously, for a function   on   with Fourier transform  , the Bochner–Riesz means of complex order  ,   (where   and  ) are defined as

 

Application to convolution operators

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For   and  ,   and   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   spaces is much simpler than the problem of "regular" almost everywhere convergence of Fourier series/integrals (corresponding to  ).

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

 

the kernel is no longer integrable. Here, establishing almost everywhere convergence becomes correspondingly more difficult.

Bochner–Riesz conjecture

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

By a transference result, the   and   problems are equivalent to one another, and as such, by an argument using the uniform boundedness principle, for any particular  ,   norm convergence follows in both cases for exactly those   where   is the symbol of an   bounded Fourier multiplier operator.

For  , that question has been completely resolved, but for  , it has only been partially answered. The case of   is not interesting here as convergence follows for   in the most difficult   case as a consequence of the   boundedness of the Hilbert transform and an argument of Marcel Riesz.

Define  , the "critical index", as

 .

Then the Bochner–Riesz conjecture states that

 

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

References

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  1. ^ Fefferman, Charles (1971). "The multiplier problem for the ball". Annals of Mathematics. 94 (2): 330–336. doi:10.2307/1970864. JSTOR 1970864.
  2. ^ Ciatti, Paolo (2008). Topics in Mathematical Analysis. World Scientific. p. 347. ISBN 9789812811066.

Further reading

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  • 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.