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:

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

where the Fourier coefficient is defined by . If , then

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

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