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.
be a periodic function, thought of as being on the n-torus,
, and having Fourier coefficients
Then the Bochner–Riesz means of complex order
δ
( δ ) > 0
) are defined as Analogously, for a function
with Fourier transform
, the Bochner–Riesz means of complex order
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.
function converge in norm.
This issue is of fundamental importance for
, since regular spherical norm convergence (again corresponding to
This was shown in a paper of 1971 by Charles Fefferman.
[1] By a transference result, the
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
bounded Fourier multiplier operator.
, that question has been completely resolved, but for
, it has only been partially answered.
is not interesting here as convergence follows for
case as a consequence of the
boundedness of the Hilbert transform and an argument of Marcel Riesz.
, the "critical index", as Then the Bochner–Riesz conjecture states that is the necessary and sufficient condition for a
bounded Fourier multiplier operator.