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''Bochner–Riesz averages''
 
''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.
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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., <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.
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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 <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:
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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:
  
<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>
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\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 <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:
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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 <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" />.
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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 <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" />.
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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 <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" />.
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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 <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" />.
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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 <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
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If $f$ is an integrable function on the torus $\mathcal{T} ^ { n }$, the Bochner–Riesz means of order $\delta > 0$ 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>
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\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 <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
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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
  
<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>
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\begin{equation*} \operatorname { lim } _ { R } S _ { R } ^ { \delta } \,f ( x ) = f ( x ) \end{equation*}
  
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" />.
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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 <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
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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
  
<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>
+
\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 <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" />.
+
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====
 
====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>
+
<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>

Latest revision as of 19:30, 21 January 2024

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 on a Euclidean space ${\bf R} ^ { n }$, with 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 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) 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 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

[a1] S. Bochner, "Summation of multiple Fourier series by spherical means" Trans. Amer. Math. Soc. , 40 (1936) pp. 175–207 Zbl 62.0293.03
[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
How to Cite This Entry:
Bochner-Riesz means. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bochner-Riesz_means&oldid=22147
This article was adapted from an original article by Mark Pinsky (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article