Bochner-Riesz means
Bochner–Riesz averages
Bochner–Riesz means can be defined and developed in different settings: multiple Fourier integrals; multiple Fourier series; other orthogonal series expansions. Below these three separate cases will be pursued, with regard to -convergence, almost-everywhere convergence, localization, and convergence or oscillation at a pre-assigned point.
A primary motivation for studying these operations lies in the fact that a general Fourier series or Fourier integral expansion can only be expected to converge in the sense of the mean square (i.e., ) norm; by inserting various smoothing and convergence factors, the convergence can often be improved to
,
, or to the almost-everywhere sense.
If is an integrable function on a Euclidean space
, with Fourier transform
, the Bochner–Riesz means of order
are defined by:
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This also can be formally written as a convolution with a kernel function. If (the critical index), then this kernel is integrable; in particular,
is a bounded operator on
,
, and
for almost every
and
. Below the critical index, one has the following results:
If and
, then
is a bounded operator on
if and only if
lies in the trapezoidal region defined by the inequalities
.
If and
, then
is a bounded operator on
if and only if
lies in the trapezoidal region defined by the inequalities
.
If and
, then
is a bounded operator on
if
lies in the triangular region defined by the inequalities
and is an unbounded operator if either
or
.
For any , in the limiting case
,
is a bounded operator on
if and only if
. If
and
has
continuous derivatives, then
provided that
. If
in an open ball centred at
, then
when
. There is also a Gibbs phenomenon for
functions which have a simple jump across a hypersurface
with respect to
. If
, then the set of accumulation points of
when
,
equals the segment with centre
and length
, where
.
If is an integrable function on the torus
, the Bochner–Riesz means of order
are defined by
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where the Fourier coefficient is defined by . If
, then
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almost everywhere if ; convergence in
holds if
and
,
. If
,
, then
uniformly for
. At the critical index, one has the following behaviour: for any open ball centred at
, there exists an
so that
in the ball and
. There exists an integrable function
for which
for almost every
. If, in addition,
is integrable and
satisfies a Dini condition (cf. also Dini criterion) at
, then
.
Bochner–Riesz means can be defined with respect to any orthonormal basis of the Hilbert space corresponding to a self-adjoint differential operator
with eigenvalues
. In this setting, the Bochner–Riesz means of order
are defined by
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In the case of multiple Hermite series corresponding to the differential operator on
, one has
and the convergence in
holds if
; almost-everywhere convergence holds if
. In the case of an arbitrary elliptic differential operator on a compact manifold, it is known that if
, then
whenever
. For second-order operators there is an
convergence theorem, provided that
and
and
.
References
[a1] | S. Bochner, "Summation of multiple Fourier series by spherical means" Trans. Amer. Math. Soc. , 40 (1936) pp. 175–207 |
[a2] | C. Fefferman, "A note on spherical summation multipliers" Israel J. Math. , 15 (1973) pp. 44–52 |
[a3] | B.I. Golubov, "On Gibb's phenomenon for Riesz spherical means of multiple Fourier integrals and Fourier series" Anal. Math. , 4 (1978) pp. 269–287 |
[a4] | B.M. Levitan, "Ueber die Summierung mehrfacher Fourierreihen und Fourierintegrale" Dokl. Akad. Nauk SSSR , 102 (1955) pp. 1073–1076 |
[a5] | E.M. Stein, "Harmonic analysis" , Princeton Univ. Press (1993) |
[a6] | S. Thangavelu, "Lectures on Hermite and Laguerre expansions" , Princeton Univ. Press (1993) |
[a7] | C. Sogge, "On the convergence of Riesz means on compact manifolds" Ann. of Math. , 126 (1987) pp. 439–447 |
Bochner-Riesz means. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bochner-Riesz_means&oldid=16536