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
(
ξ
)
+
=
{
ξ
,
if
ξ
>
0
0
,
otherwise
.
Let
f
be a periodic function, thought of as being on the n-torus,
T
n
, and having Fourier coefficients
f
^
(
k
)
for
k
∈
Z
n
. Then the Bochner–Riesz means of complex order
δ
,
B
R
δ
f
of (where
R
>
0
and
Re
(
δ
)
>
0
) are defined as
B
R
δ
f
(
θ
)
=
∑
k
∈
Z
n
|
k
|
≤
R
(
1
−
|
k
|
2
R
2
)
+
δ
f
^
(
k
)
e
2
π
i
k
⋅
θ
.
Analogously, for a function
f
on
R
n
with Fourier transform
f
^
(
ξ
)
, the Bochner–Riesz means of complex order
δ
,
S
R
δ
f
(where
R
>
0
and
Re
(
δ
)
>
0
) are defined as
S
R
δ
f
(
x
)
=
∫
|
ξ
|
≤
R
(
1
−
|
ξ
|
2
R
2
)
+
δ
f
^
(
ξ
)
e
2
π
i
x
⋅
ξ
d
ξ
.
For
δ
>
0
and
n
=
1
,
S
R
δ
and
B
R
δ
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
δ
=
0
). In higher dimensions, the convolution kernels become more "badly behaved" (specifically, for
δ
≤
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
δ
and which
p
the Bochner–Riesz means of an
L
p
function converge in norm. This is of fundamental importance for
n
≥
2
, since regular spherical norm convergence (again corresponding to
δ
=
0
) fails in
L
p
when
p
≠
2
. This was shown in a paper of 1971 by Charles Fefferman. 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
∈
(
1
,
∞
)
,
L
p
norm convergence follows in both cases for exactly those
δ
where
(
1
−
|
ξ
|
2
)
+
δ
is the symbol of an
L
p
bounded Fourier multiplier operator. For
n
=
2
, this question has been completely resolved, but for
n
≥
3
, 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
L
p
boundedness of the Hilbert transform and an argument of Marcel Riesz.
