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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 $L ^ { p }$-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., $L^{2}$) norm; by inserting various smoothing and convergence factors, the convergence can often be improved to $L ^ { p }$, $p \neq 2$, or to the almost-everywhere sense.

If $f$ is an [[Integrable function|integrable function]] on a Euclidean space ${\bf R} ^ { n }$, with [[Fourier transform|Fourier transform]] $\hat { f } ( \xi ) = \int _ { \mathbf{R} ^ { n } } f ( x ) e ^ { - 2 \pi i x . \xi } d x$, the Bochner–Riesz means of order $\delta > 0$ are defined by:

\begin{equation*} M _ { R } ^ { \delta } (\, f ) ( x ) = \int _ { | \xi | \leq R } \left( 1 - \frac { | \xi | ^ { 2 } } { R ^ { 2 } } \right) ^ { \delta } e ^ { 2 \pi i x \cdot \xi } \hat { f } ( \xi ) d \xi . \end{equation*}

This also can be formally written as a convolution with a kernel function. If $\delta > ( n - 1 ) / 2$ (the critical index), then this kernel is integrable; in particular, $M _ { R } ^ { \delta }$ is a bounded operator on $L ^ { p } ( \mathbf{R} ^ { n } )$, $1 \leq p < \infty$, and $\operatorname { lim } _ { R \rightarrow \infty } M _ { R } ^ { \delta } f ( x ) = f ( x )$ for almost every $x \in \mathbf{R} ^ { n }$ and $\| M _ { R } ^ { \delta } f - f \| _ { p } \rightarrow 0$. Below the critical index, one has the following results:

If $n = 2$ and $0 < \delta \leq 1 / 2$, then $M _ { R } ^ { \delta }$ is a bounded operator on $L ^ { p }$ if and only if $( 1 / p , \delta )$ lies in the trapezoidal region defined by the inequalities $( 1 - 2 \delta ) / 4 < 1 / p < ( 3 + 2 \delta ) / 4$.

If $n \geq 3$ and $( n - 1 ) / 2 ( n + 1 ) < \delta < ( n - 1 ) / 2$, then $M _ { R } ^ { \delta }$ is a bounded operator on $L ^ { p }$ if and only if $( 1 / p , \delta )$ lies in the trapezoidal region defined by the inequalities $( n - 1 - 2 \delta ) / 2 n < 1 / p < ( n + 1 + 2 \delta ) / 2 n$.

If $n \geq 3$ and $0 \leq \delta \leq ( n - 1 ) / 2 ( n + 1 )$, then $M _ { R } ^ { \delta }$ is a bounded operator on $L ^ { p }$ if $( 1 / p , \delta )$ lies in the triangular region defined by the inequalities $( n - 1 - 2 \delta ) / 2 n < 1 / p < ( n - 1 + 2 \delta ) / 2 n$ and is an unbounded operator if either $1 / p \leq ( n - 1 - 2 \delta ) / 2 n$ or $1 / p \geq ( n + 1 + 2 \delta ) / 2 n$.

For any $n \geq 2$, in the limiting case $\delta = 0$, $M _ { R } ^ { \delta }$ is a bounded operator on $L ^ { p }$ if and only if $p = 2$. If $f \in L ^ { 1 } \cap L ^ { 2 } ( \mathbf{R} ^ { 2 k + 1 } )$ and $f$ has $j$ continuous derivatives, then $\operatorname { lim } _ { R } M _ { R } ^ { \delta } f ( x ) = f ( x )$ provided that $\delta \geq k - j$. If $f = 0$ in an open ball centred at $0$, then $M _ { R } ^ { ( n - 1 ) / 2 } f ( 0 ) \rightarrow 0$ when $R \rightarrow \infty$. There is also a [[Gibbs phenomenon|Gibbs phenomenon]] for $L^1$ functions which have a simple jump across a hypersurface $S$ with respect to $x _ { 0 } \in S$. If $\delta > ( n - 1 ) / 2$, then the set of accumulation points of $M _ { R } f ( x )$ when $R \rightarrow \infty$, $x \rightarrow x_{0}$ equals the segment with centre $[ f _ { S } ^ { + } ( x _ { 0 } ) + f _ { S } ^ { - } ( x _ { 0 } ) ] / 2$ and length $G _ { \delta } [ f _ { S } ^ { + } ( x _ { 0 } ) - f _ { S } ^ { - } ( x _ { 0 } ) ]$, where $G _ { \delta } = ( 2 / \pi ) \operatorname { sup } _ { x > 0 } \int _ { 0 } ^ { 1 } ( 1 - t ^ { 2 } ) ^ { \delta } \operatorname { sin } x t d t / t$.

If $f$ is an integrable function on the torus $\mathcal{T} ^ { n }$, the Bochner–Riesz means of order $\delta > 0$ are defined by

\begin{equation*} S _ { R } ^ { \delta } ( f ) ( x ) = \sum _ { | m | \leq R } \left( 1 - \frac { | m | ^ { 2 } } { R ^ { 2 } } \right) ^ { \delta } e ^ { 2 \pi i x m } \hat { f } ( m ), \end{equation*}

where the Fourier coefficient is defined by $\widehat { f } ( m ) = \int _ { \mathcal T ^ { n } } f ( x ) e ^ { - 2 \pi i x m } d x$. If $f \in L ^ { p } ( \mathcal{T} ^ { n } )$, then

\begin{equation*} \operatorname { lim } _ { R } S _ { R } ^ { \delta } \,f ( x ) = f ( x ) \end{equation*}

almost everywhere if $\delta > ( n - 1 ) | 1 / 2 - 1 / p |$; convergence in $L ^ { p }$ holds if $| 1 / p - 1 / 2 | \geq 1 / ( n + 1 )$ and $\delta > 0$, $\delta > | ( 1 / n p ) - ( 1 / 2 n ) | - 1 / 2$. If $f \in C ( \mathcal{T} ^ { n } )$, $\delta > ( n - 1 ) / 2$, then $\operatorname{lim}S _ { R } ^ { \delta } ( x ) = f ( x )$ uniformly for $x \in \mathcal{T} ^ { n }$. At the critical index, one has the following behaviour: for any open ball centred at $0$, there exists an $f \in L ^ { 1 } ( \mathcal{T} ^ { n } )$ so that $f = 0$ in the ball and $\operatorname{lim\,sup}_R S _ { R } ^ { ( n - 1 ) / 2 } f ( 0 ) = + \infty$. There exists an integrable function $f$ for which $\operatorname{lim sup}_R S _ { R } ^ { ( n - 1 ) / 2 } f ( x ) = + \infty$ for almost every $x \in \mathcal{T} ^ { n }$. If, in addition, $| f | \operatorname { log } ^ { + } | f |$ is integrable and $f$ satisfies a Dini condition (cf. also [[Dini criterion|Dini criterion]]) at $x _ { 0 }$, then $\lim _R  S _ { R } ^ { ( n - 1 ) / 2 } f ( x _ { 0 } ) = f ( x _ { 0 } )$.

Bochner–Riesz means can be defined with respect to any orthonormal basis $\{ \phi _ { k } \}$ of the [[Hilbert space|Hilbert space]] corresponding to a self-adjoint differential operator $L$ with eigenvalues $\lambda _ { k } \geq 0$. In this setting, the Bochner–Riesz means of order $\delta > 0$ are defined by

\begin{equation*} S _ { R } ^ { \delta }\, f ( x ) = \sum _ { \lambda _ { k } \leq R } \left( 1 - \frac { \lambda _ { k } } { R } \right) ^ { \delta } ( f , \phi _ { k } ) \phi _ { k } ( x ). \end{equation*}

In the case of multiple Hermite series corresponding to the differential operator $L = ( \Delta / 2 ) - x . \nabla$ on ${\bf R} ^ { n }$, one has $\lambda _ { k } = 2 k + n$ and the convergence in $L ^ { p }$ holds if $\delta > ( n - 1 ) / 2$; almost-everywhere convergence holds if $\delta > ( 3 n - 2 ) / 6$. In the case of an arbitrary elliptic differential operator on a compact manifold, it is known that if $f \in L ^ { 1 }$, then $\| S _ { R } ^ { \delta }\, f - f \| _ { 1 } \rightarrow 0$ whenever $\delta > ( n - 1 ) / 2$. For second-order operators there is an $L ^ { p }$ convergence theorem, provided that $| 1 / p - 1 / 2 | \geq 1 / ( n + 1 )$ and $\delta > 0$ and $\delta > | 1 / n p - 1 / 2 n | - 1 / 2$.

====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 {{ZBL|62.0293.03}}</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>