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''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b1203102.png" />-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., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b1203103.png" />) norm; by inserting various smoothing and convergence factors, the convergence can often be improved to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b1203104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b1203105.png" />, or to the almost-everywhere sense.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b1203106.png" /> is an [[Integrable function|integrable function]] on a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b1203107.png" />, with [[Fourier transform|Fourier transform]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b1203108.png" />, the Bochner–Riesz means of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031010.png" /> are defined by:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031011.png" /></td> </tr></table>

This also can be formally written as a convolution with a kernel function. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031012.png" /> (the critical index), then this kernel is integrable; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031013.png" /> is a bounded operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031016.png" /> for almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031018.png" />. Below the critical index, one has the following results:

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031021.png" /> is a bounded operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031022.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031023.png" /> lies in the trapezoidal region defined by the inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031024.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031027.png" /> is a bounded operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031028.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031029.png" /> lies in the trapezoidal region defined by the inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031030.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031033.png" /> is a bounded operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031034.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031035.png" /> lies in the triangular region defined by the inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031036.png" /> and is an unbounded operator if either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031037.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031038.png" />.

For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031039.png" />, in the limiting case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031041.png" /> is a bounded operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031042.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031045.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031046.png" /> continuous derivatives, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031047.png" /> provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031048.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031049.png" /> in an open ball centred at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031051.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031052.png" />. There is also a [[Gibbs phenomenon|Gibbs phenomenon]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031053.png" /> functions which have a simple jump across a hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031054.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031055.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031056.png" />, then the set of accumulation points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031057.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031059.png" /> equals the segment with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031060.png" /> and length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031061.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031062.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031063.png" /> is an integrable function on the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031064.png" />, the Bochner–Riesz means of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031065.png" /> are defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031066.png" /></td> </tr></table>

where the Fourier coefficient is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031067.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031068.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031069.png" /></td> </tr></table>

almost everywhere if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031070.png" />; convergence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031071.png" /> holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031074.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031076.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031077.png" /> uniformly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031078.png" />. At the critical index, one has the following behaviour: for any open ball centred at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031079.png" />, there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031080.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031081.png" /> in the ball and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031082.png" />. There exists an integrable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031083.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031084.png" /> for almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031085.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031086.png" /> is integrable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031087.png" /> satisfies a Dini condition (cf. also [[Dini criterion|Dini criterion]]) at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031088.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031089.png" />.

Bochner–Riesz means can be defined with respect to any orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031090.png" /> of the [[Hilbert space|Hilbert space]] corresponding to a self-adjoint differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031091.png" /> with eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031092.png" />. In this setting, the Bochner–Riesz means of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031093.png" /> are defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031094.png" /></td> </tr></table>

In the case of multiple Hermite series corresponding to the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031095.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031096.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031097.png" /> and the convergence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031098.png" /> holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b12031099.png" />; almost-everywhere convergence holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b120310100.png" />. In the case of an arbitrary elliptic differential operator on a compact manifold, it is known that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b120310101.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b120310102.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b120310103.png" />. For second-order operators there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b120310104.png" /> convergence theorem, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b120310105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b120310106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120310/b120310107.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bochner,   "Summation of multiple Fourier series by spherical means"  ''Trans. Amer. Math. Soc.'' , '''40'''  (1936)  pp. 175–207</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Fefferman,   "A note on spherical summation multipliers"  ''Israel J. Math.'' , '''15'''  (1973)  pp. 44–52</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B.M. Levitan,   "Ueber die Summierung mehrfacher Fourierreihen und Fourierintegrale"  ''Dokl. Akad. Nauk SSSR'' , '''102'''  (1955)  pp. 1073–1076</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.M. Stein,   "Harmonic analysis" , Princeton Univ. Press  (1993)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Thangavelu,   "Lectures on Hermite and Laguerre expansions" , Princeton Univ. Press  (1993)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C. Sogge,   "On the convergence of Riesz means on compact manifolds"  ''Ann. of Math.'' , '''126'''  (1987)  pp. 439–447</TD></TR></table>