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A formula giving an integral representation of the identity operator and having a number of realizations depending on the initial setting. Historically, it is usually connected with the paper [[#References|[a1]]] by A.P. Calderón (1964), but its basic idea was known before.
 
A formula giving an integral representation of the identity operator and having a number of realizations depending on the initial setting. Historically, it is usually connected with the paper [[#References|[a1]]] by A.P. Calderón (1964), but its basic idea was known before.
  
 
The Calderón reproducing formula is widely used in the theory of continuous wavelet transforms [[#References|[a2]]] (cf. also [[Wavelet analysis|Wavelet analysis]]).
 
The Calderón reproducing formula is widely used in the theory of continuous wavelet transforms [[#References|[a2]]] (cf. also [[Wavelet analysis|Wavelet analysis]]).
  
Given a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c1200201.png" /> and sufficiently nice radial functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c1200202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c1200203.png" />, the Calderón reproducing formula reads:
+
Given a function $f \in L ^ { 2 } ( \mathbf{R} ^ { n } )$ and sufficiently nice radial functions $u$ and $v$, the Calderón reproducing formula reads:
  
<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/c/c120/c120020/c1200204.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation} \tag{a1} \int _ { 0 } ^ { \infty } \frac { f  *  u _ { t }  *  v _ { t } } { t } d t = c _ { u , v }\, f, \end{equation}
  
<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/c/c120/c120020/c1200205.png" /></td> </tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c1200205.png"/></td> </tr></table>
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c1200206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c1200207.png" />,  "*"  is a convolution operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c1200208.png" /> is the area of the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c1200209.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002012.png" /> designate the Fourier transforms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002014.png" /> (cf. also [[Fourier transform|Fourier transform]]). The convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002015.png" /> is called the continuous wavelet transform, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002016.png" /> is called an analyzing wavelet and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002017.png" /> is called a reconstructing wavelet. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002018.png" /> is called admissible if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002019.png" />. The integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002020.png" /> in (a1) is treated as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002021.png" />-limit of the corresponding truncated integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002022.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002024.png" />.
+
Here, $u _ { t } ( x ) = t ^ { - n } u ( x / t )$, $v _ { t } ( x ) = t ^ { - n } v ( x / t )$,  "*"  is a convolution operator, $| S ^ { n - 1 } |$ is the area of the unit sphere $S ^ { n - 1 }$ in ${\bf R} ^ { n }$, and $\widehat{u}$ and $\hat{v} $ designate the Fourier transforms of $u$ and $v$ (cf. also [[Fourier transform|Fourier transform]]). The convolution $( W _ { u } f ) ( x , t ) = ( f ^ { * } u _ { t } ) ( x )$ is called the continuous wavelet transform, $u ( x )$ is called an analyzing wavelet and $v ( x )$ is called a reconstructing wavelet. The pair $( u , v )$ is called admissible if $0 \neq \mathfrak { c } _ { u , v}  &lt; \infty$. The integral $\int _ { 0 } ^ { \infty }$ in (a1) is treated as the $L^{2}$-limit of the corresponding truncated integral $\int _ { \epsilon } ^ { \rho }$ as $\epsilon \rightarrow 0$ and $\rho \rightarrow \infty$.
  
 
A generalization of (a1) has the form [[#References|[a3]]]:
 
A generalization of (a1) has the form [[#References|[a3]]]:
  
<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/c/c120/c120020/c12002025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} \int _ { 0 } ^ { \infty } \frac { f ^ {*}  \mu _ { t } } { t } d t \equiv \operatorname { lim } _ { \epsilon \rightarrow 0 , \rho \rightarrow \infty } \int _ { \epsilon } ^ { \rho } \frac { f ^ {*}  \mu _ { t } } { t } d t = c _ { \mu } \,f, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002026.png" /> is a suitable radial [[Borel measure|Borel measure]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002027.png" /> stands for the dilation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002028.png" />, and the limit is interpreted in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002029.png" />-norm and in the  "almost everywhere"  sense. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002030.png" /> is not radial, then the left-hand side of (a2) is a sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002032.png" /> is a certain Calderón–Zygmund singular integral (cf. also [[Singular integral|Singular integral]]; [[#References|[a4]]]). At many occurrences one can write
+
where $\mu$ is a suitable radial [[Borel measure|Borel measure]], $\mu _ { t }$ stands for the dilation of $\mu$, and the limit is interpreted in the $L ^ { p }$-norm and in the  "almost everywhere"  sense. If $\mu$ is not radial, then the left-hand side of (a2) is a sum $c _ { \mu } f + T _ { \mu } f$, where $T _ { \mu } f$ is a certain Calderón–Zygmund singular integral (cf. also [[Singular integral|Singular integral]]; [[#References|[a4]]]). At many occurrences one can write
  
<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/c/c120/c120020/c12002033.png" /></td> </tr></table>
+
\begin{equation*} \mu _ { t } = t \frac { \partial } { \partial t } k _ { t }, \end{equation*}
  
provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002034.png" /> is an approximate identity (cf. also [[Involution algebra|Involution algebra]]). A formal integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002035.png" /> (cf. (a2)) can be regarded as an integral representation of the delta-function. More general representations, corresponding to inhomogeneous dilations on homogeneous groups, are given in [[#References|[a5]]].
+
provided that $k _ { t } ^ { * } f$ is an approximate identity (cf. also [[Involution algebra|Involution algebra]]). A formal integral $\int _ { 0 } ^ { \infty } \mu _ { t } d t / t$ (cf. (a2)) can be regarded as an integral representation of the delta-function. More general representations, corresponding to inhomogeneous dilations on homogeneous groups, are given in [[#References|[a5]]].
  
Calderón-type reproducing formulas (and the relevant continuous wavelet transforms) can be obtained starting out from analytic families of operators, involving the identity operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002036.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002037.png" /> is such a family, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002038.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002039.png" /> is an [[Analytic continuation|analytic continuation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002040.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002041.png" />-variable, then the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002042.png" /> can be written as a Calderón-type reproducing formula provided that a suitable notion of analytic continuation is employed. For this purpose one can use a generalization of the method of A. Marchaud, described in [[#References|[a6]]], Sec. 10.7. In accordance with the generalized Marchaud method, (a2) can be related to the analytic family of Riesz potentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002043.png" /> (cf. also [[Riesz potential|Riesz potential]]) defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002044.png" />, and possessing a wavelet-type representation of the form
+
Calderón-type reproducing formulas (and the relevant continuous wavelet transforms) can be obtained starting out from analytic families of operators, involving the identity operator $I$. If $\{ A ^ { \alpha } \}$ is such a family, $A ^ { 0 } = I$, and $\operatorname{a.c.}A ^ { \alpha } f$ is an [[Analytic continuation|analytic continuation]] of $A ^ { \alpha } f$ in the $\alpha$-variable, then the equality $( \text { a.c. } A ^ { \alpha } f ) _ { \alpha = 0 } = f$ can be written as a Calderón-type reproducing formula provided that a suitable notion of analytic continuation is employed. For this purpose one can use a generalization of the method of A. Marchaud, described in [[#References|[a6]]], Sec. 10.7. In accordance with the generalized Marchaud method, (a2) can be related to the analytic family of Riesz potentials $I ^ { \alpha } f$ (cf. also [[Riesz potential|Riesz potential]]) defined by $\widehat { ( I ^ { \alpha } f ) } ( \xi ) = | \xi | ^ { - \alpha } \hat { f } ( \xi )$, and possessing a wavelet-type representation of the form
  
<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/c/c120/c120020/c12002045.png" /></td> </tr></table>
+
\begin{equation*} ( I ^ { \alpha } f ) ( x ) = c _ { \mu , \alpha } \int _ { 0 } ^ { \infty } ( f ^ { * } \mu _ { t } ) ( x ) t ^ { \alpha - 1 } d t, \end{equation*}
  
cf. [[#References|[a6]]], Sec. 17. An analogue of (a2) for the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002046.png" />, corresponding to the spherical Riesz potentials, reads [[#References|[a6]]], Sec. 34:
+
cf. [[#References|[a6]]], Sec. 17. An analogue of (a2) for the unit sphere $S ^ { 2 }$, corresponding to the spherical Riesz potentials, reads [[#References|[a6]]], Sec. 34:
  
<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/c/c120/c120020/c12002047.png" /></td> </tr></table>
+
\begin{equation*} \int _ { 0 } ^ { \infty } ( V _ { g } f ) ( \theta , t ) \frac { d t } { t } = c _ { g } \,f, \end{equation*}
  
<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/c/c120/c120020/c12002048.png" /></td> </tr></table>
+
\begin{equation*}  c _g = \int _ { 0 } ^ { \infty } g ( t ) \operatorname { log } \frac { 1 } { t } d t, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002049.png" /> and
+
where $\theta \in S ^ { 2 }$ and
  
<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/c/c120/c120020/c12002050.png" /></td> </tr></table>
+
\begin{equation*} ( V _ { g } f ) ( \theta , t ) = ( 2 \pi t ) ^ { - 1 } \int _ { S ^ { 2 } } f ( \sigma ) g \left( \frac { 1 - \theta . \sigma } { t } \right) d \sigma \end{equation*}
  
 
is a spherical convolution, which can be regarded as a  "spherical wavelet transform" .
 
is a spherical convolution, which can be regarded as a  "spherical wavelet transform" .
  
Further generalizations of (a2) can be associated with integral representations of the unit mass uniformly distributed on a sufficiently smooth surface. In such a case, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002051.png" /> on the right-hand side of (a2) should be replaced by the relevant [[Radon transform|Radon transform]]. Some examples related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002053.png" />-plane transforms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002054.png" /> and spherical Radon transforms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002055.png" /> can be found in [[#References|[a7]]].
+
Further generalizations of (a2) can be associated with integral representations of the unit mass uniformly distributed on a sufficiently smooth surface. In such a case, the function $f$ on the right-hand side of (a2) should be replaced by the relevant [[Radon transform|Radon transform]]. Some examples related to $k$-plane transforms in ${\bf R} ^ { n }$ and spherical Radon transforms in $S ^ { n }$ can be found in [[#References|[a7]]].
  
Formula (a2) can be extended to non-radial measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002056.png" /> as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002057.png" /> be the special [[Orthogonal group|orthogonal group]] of rotations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002058.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002060.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002061.png" /> be the rotated and dilated version of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002062.png" />. A natural generalization of (a2) reads [[#References|[a3]]]:
+
Formula (a2) can be extended to non-radial measures $\mu$ as follows. Let $\operatorname{SO} ( n )$ be the special [[Orthogonal group|orthogonal group]] of rotations of ${\bf R} ^ { n }$. For $\gamma \in \operatorname{SO} ( n )$ and $t &gt; 0$, let $\mu_{ \gamma , t}$ be the rotated and dilated version of $\mu$. A natural generalization of (a2) reads [[#References|[a3]]]:
  
<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/c/c120/c120020/c12002063.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} \int _ { \operatorname{SO} ( n ) } d \gamma \int _ { 0 } ^ { \infty } \frac { f ^ { * } \mu _ { \gamma , t } } { t } d t = c _ { \mu } f. \end{equation}
  
A remarkable feature of this formula is that, with a suitable choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002064.png" />, it gives rise to a series of explicit inversion formulas for a number of important transforms in [[Integral geometry|integral geometry]] [[#References|[a3]]]. For example, assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002065.png" /> is represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002069.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002070.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002071.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002072.png" /> is the Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002074.png" /> is a suitable reconstructing measure on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002075.png" />-plane. Then (a3) can be rewritten as an inversion formula for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002077.png" />-plane transform, which assigns to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002078.png" /> a collection of integrals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002079.png" /> over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002080.png" />-dimensional planes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002081.png" />. The idea of such an application of the Calderón reproducing formula is due to M. Holschneider [[#References|[a8]]], Sec. 12, who considered the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002082.png" />. Another choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002083.png" /> in (a3) leads to an explicit inversion of windowed X-ray transforms (cf. also [[X-ray transform|X-ray transform]]), defined by
+
A remarkable feature of this formula is that, with a suitable choice of $\mu$, it gives rise to a series of explicit inversion formulas for a number of important transforms in [[Integral geometry|integral geometry]] [[#References|[a3]]]. For example, assume that $x \in \mathbf{R} ^ { n }$ is represented as $x = ( x ^ { \prime } , x ^ { \prime \prime } )$, $x ^ { \prime } = ( x _ { 1 } , \dots , x _ { k } )$, $x ^ { \prime \prime } = ( x _ { k+1}, \dots , x _ { n } )$, $1 \leq k \leq n - 1$, and $\mu \equiv \mu ( x )$ has the form $\mu ( x ) = m ( x ^ { \prime } ) \times \lambda ( x ^ { \prime \prime } )$, where $m ( x ^ { \prime } )$ is the Lebesgue measure on $\mathbf{R} ^ { k }$ and $\lambda ( x ^ { \prime \prime } )$ is a suitable reconstructing measure on the $x ^ { \prime \prime }$-plane. Then (a3) can be rewritten as an inversion formula for the $k$-plane transform, which assigns to $f$ a collection of integrals of $f$ over all $k$-dimensional planes in ${\bf R} ^ { n }$. The idea of such an application of the Calderón reproducing formula is due to M. Holschneider [[#References|[a8]]], Sec. 12, who considered the case $n = 2$. Another choice of $\mu$ in (a3) leads to an explicit inversion of windowed X-ray transforms (cf. also [[X-ray transform|X-ray transform]]), 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/c/c120/c120020/c12002084.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a4} ( X _ { \nu } f ) ( x , y ) = \int _ { - \infty } ^ { \infty } f ( x + t y ) d \nu ( t ), \end{equation}
  
<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/c/c120/c120020/c12002085.png" /></td> </tr></table>
+
\begin{equation*} x , y \in \mathbf{R} ^ { n } \end{equation*}
  
with a finite Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002086.png" />. The transform (a4) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002087.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002088.png" /> is a  "nice"  compactly supported function (a window function), was introduced by G.A. Kaizer and R.F. Streater [[#References|[a9]]] in connection with applications in physics.
+
with a finite Borel measure $\nu$. The transform (a4) with $d \nu ( t ) = g ( t ) d t$, where $g$ is a  "nice"  compactly supported function (a window function), was introduced by G.A. Kaizer and R.F. Streater [[#References|[a9]]] in connection with applications in physics.
  
 
The Calderón reproducing formula admits various discrete versions, which serve as natural analogues of atomic decompositions and play an important role in the study of function spaces [[#References|[a10]]].
 
The Calderón reproducing formula admits various discrete versions, which serve as natural analogues of atomic decompositions and play an important role in the study of function spaces [[#References|[a10]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.P. Calderón,  "Intermediate spaces and interpolation, the complex method"  ''Studia Math.'' , '''24'''  (1964)  pp. 113–190</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Y. Meyer,  "Wavelets and operators" , Cambridge Univ. Press  (1992)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Rubin,  "The Calderón reproducing formula, windowed X-ray transforms and Radon transforms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120020/c12002089.png" />-spaces"  ''J. Fourier Anal. Appl.'' , '''4'''  (1998)  pp. 175–197</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D. Ryabogin,  B. Rubin,  "Singular integral operators generated by wavelet transforms"  ''Integral Eq. Operator Th.''  (in press},)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G.B. Folland,  E.M. Stein,  "Hardy spaces on homogeneous groups" , Princeton Univ. Press  (1982)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Rubin,  "Fractional integrals and potentials" , Addison-Wesley  (1996)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  B. Rubin,  "Fractional calculus and wavelet transforms in integral geometry"  ''Fractional Calculus and Applied Analysis'' , '''2'''  (1998)  pp. 193–219</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. Holschneider,  "Wavelets: an analysis tool" , Clarendon Press  (1995)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G.A. Kaiser,  R.F. Streater,  "Windowed Radon transforms, analytic signals, and wave equation"  C.K. Chui (ed.) , ''Wavelets: A Tutorial in Theory and Applications'' , Acad. Press  (1992)  pp. 399–441</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  M. Frazier,  B. Jawerth,  G. Weiss,  "Littlewood–Paley theory and the study of function spaces" , ''CBMS Reg. Conf. Ser. Math.'' , '''79''' , Amer. Math. Soc.  (1991)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  A.P. Calderón,  "Intermediate spaces and interpolation, the complex method"  ''Studia Math.'' , '''24'''  (1964)  pp. 113–190</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  Y. Meyer,  "Wavelets and operators" , Cambridge Univ. Press  (1992)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  B. Rubin,  "The Calderón reproducing formula, windowed X-ray transforms and Radon transforms in $L ^ { p }$-spaces"  ''J. Fourier Anal. Appl.'' , '''4'''  (1998)  pp. 175–197</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  D. Ryabogin,  B. Rubin,  "Singular integral operators generated by wavelet transforms"  ''Integral Eq. Operator Th.''  (in press},)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  G.B. Folland,  E.M. Stein,  "Hardy spaces on homogeneous groups" , Princeton Univ. Press  (1982)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  B. Rubin,  "Fractional integrals and potentials" , Addison-Wesley  (1996)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  B. Rubin,  "Fractional calculus and wavelet transforms in integral geometry"  ''Fractional Calculus and Applied Analysis'' , '''2'''  (1998)  pp. 193–219</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  M. Holschneider,  "Wavelets: an analysis tool" , Clarendon Press  (1995)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  G.A. Kaiser,  R.F. Streater,  "Windowed Radon transforms, analytic signals, and wave equation"  C.K. Chui (ed.) , ''Wavelets: A Tutorial in Theory and Applications'' , Acad. Press  (1992)  pp. 399–441</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  M. Frazier,  B. Jawerth,  G. Weiss,  "Littlewood–Paley theory and the study of function spaces" , ''CBMS Reg. Conf. Ser. Math.'' , '''79''' , Amer. Math. Soc.  (1991)</td></tr></table>

Latest revision as of 17:45, 1 July 2020

A formula giving an integral representation of the identity operator and having a number of realizations depending on the initial setting. Historically, it is usually connected with the paper [a1] by A.P. Calderón (1964), but its basic idea was known before.

The Calderón reproducing formula is widely used in the theory of continuous wavelet transforms [a2] (cf. also Wavelet analysis).

Given a function $f \in L ^ { 2 } ( \mathbf{R} ^ { n } )$ and sufficiently nice radial functions $u$ and $v$, the Calderón reproducing formula reads:

\begin{equation} \tag{a1} \int _ { 0 } ^ { \infty } \frac { f * u _ { t } * v _ { t } } { t } d t = c _ { u , v }\, f, \end{equation}

Here, $u _ { t } ( x ) = t ^ { - n } u ( x / t )$, $v _ { t } ( x ) = t ^ { - n } v ( x / t )$, "*" is a convolution operator, $| S ^ { n - 1 } |$ is the area of the unit sphere $S ^ { n - 1 }$ in ${\bf R} ^ { n }$, and $\widehat{u}$ and $\hat{v} $ designate the Fourier transforms of $u$ and $v$ (cf. also Fourier transform). The convolution $( W _ { u } f ) ( x , t ) = ( f ^ { * } u _ { t } ) ( x )$ is called the continuous wavelet transform, $u ( x )$ is called an analyzing wavelet and $v ( x )$ is called a reconstructing wavelet. The pair $( u , v )$ is called admissible if $0 \neq \mathfrak { c } _ { u , v} < \infty$. The integral $\int _ { 0 } ^ { \infty }$ in (a1) is treated as the $L^{2}$-limit of the corresponding truncated integral $\int _ { \epsilon } ^ { \rho }$ as $\epsilon \rightarrow 0$ and $\rho \rightarrow \infty$.

A generalization of (a1) has the form [a3]:

\begin{equation} \tag{a2} \int _ { 0 } ^ { \infty } \frac { f ^ {*} \mu _ { t } } { t } d t \equiv \operatorname { lim } _ { \epsilon \rightarrow 0 , \rho \rightarrow \infty } \int _ { \epsilon } ^ { \rho } \frac { f ^ {*} \mu _ { t } } { t } d t = c _ { \mu } \,f, \end{equation}

where $\mu$ is a suitable radial Borel measure, $\mu _ { t }$ stands for the dilation of $\mu$, and the limit is interpreted in the $L ^ { p }$-norm and in the "almost everywhere" sense. If $\mu$ is not radial, then the left-hand side of (a2) is a sum $c _ { \mu } f + T _ { \mu } f$, where $T _ { \mu } f$ is a certain Calderón–Zygmund singular integral (cf. also Singular integral; [a4]). At many occurrences one can write

\begin{equation*} \mu _ { t } = t \frac { \partial } { \partial t } k _ { t }, \end{equation*}

provided that $k _ { t } ^ { * } f$ is an approximate identity (cf. also Involution algebra). A formal integral $\int _ { 0 } ^ { \infty } \mu _ { t } d t / t$ (cf. (a2)) can be regarded as an integral representation of the delta-function. More general representations, corresponding to inhomogeneous dilations on homogeneous groups, are given in [a5].

Calderón-type reproducing formulas (and the relevant continuous wavelet transforms) can be obtained starting out from analytic families of operators, involving the identity operator $I$. If $\{ A ^ { \alpha } \}$ is such a family, $A ^ { 0 } = I$, and $\operatorname{a.c.}A ^ { \alpha } f$ is an analytic continuation of $A ^ { \alpha } f$ in the $\alpha$-variable, then the equality $( \text { a.c. } A ^ { \alpha } f ) _ { \alpha = 0 } = f$ can be written as a Calderón-type reproducing formula provided that a suitable notion of analytic continuation is employed. For this purpose one can use a generalization of the method of A. Marchaud, described in [a6], Sec. 10.7. In accordance with the generalized Marchaud method, (a2) can be related to the analytic family of Riesz potentials $I ^ { \alpha } f$ (cf. also Riesz potential) defined by $\widehat { ( I ^ { \alpha } f ) } ( \xi ) = | \xi | ^ { - \alpha } \hat { f } ( \xi )$, and possessing a wavelet-type representation of the form

\begin{equation*} ( I ^ { \alpha } f ) ( x ) = c _ { \mu , \alpha } \int _ { 0 } ^ { \infty } ( f ^ { * } \mu _ { t } ) ( x ) t ^ { \alpha - 1 } d t, \end{equation*}

cf. [a6], Sec. 17. An analogue of (a2) for the unit sphere $S ^ { 2 }$, corresponding to the spherical Riesz potentials, reads [a6], Sec. 34:

\begin{equation*} \int _ { 0 } ^ { \infty } ( V _ { g } f ) ( \theta , t ) \frac { d t } { t } = c _ { g } \,f, \end{equation*}

\begin{equation*} c _g = \int _ { 0 } ^ { \infty } g ( t ) \operatorname { log } \frac { 1 } { t } d t, \end{equation*}

where $\theta \in S ^ { 2 }$ and

\begin{equation*} ( V _ { g } f ) ( \theta , t ) = ( 2 \pi t ) ^ { - 1 } \int _ { S ^ { 2 } } f ( \sigma ) g \left( \frac { 1 - \theta . \sigma } { t } \right) d \sigma \end{equation*}

is a spherical convolution, which can be regarded as a "spherical wavelet transform" .

Further generalizations of (a2) can be associated with integral representations of the unit mass uniformly distributed on a sufficiently smooth surface. In such a case, the function $f$ on the right-hand side of (a2) should be replaced by the relevant Radon transform. Some examples related to $k$-plane transforms in ${\bf R} ^ { n }$ and spherical Radon transforms in $S ^ { n }$ can be found in [a7].

Formula (a2) can be extended to non-radial measures $\mu$ as follows. Let $\operatorname{SO} ( n )$ be the special orthogonal group of rotations of ${\bf R} ^ { n }$. For $\gamma \in \operatorname{SO} ( n )$ and $t > 0$, let $\mu_{ \gamma , t}$ be the rotated and dilated version of $\mu$. A natural generalization of (a2) reads [a3]:

\begin{equation} \tag{a3} \int _ { \operatorname{SO} ( n ) } d \gamma \int _ { 0 } ^ { \infty } \frac { f ^ { * } \mu _ { \gamma , t } } { t } d t = c _ { \mu } f. \end{equation}

A remarkable feature of this formula is that, with a suitable choice of $\mu$, it gives rise to a series of explicit inversion formulas for a number of important transforms in integral geometry [a3]. For example, assume that $x \in \mathbf{R} ^ { n }$ is represented as $x = ( x ^ { \prime } , x ^ { \prime \prime } )$, $x ^ { \prime } = ( x _ { 1 } , \dots , x _ { k } )$, $x ^ { \prime \prime } = ( x _ { k+1}, \dots , x _ { n } )$, $1 \leq k \leq n - 1$, and $\mu \equiv \mu ( x )$ has the form $\mu ( x ) = m ( x ^ { \prime } ) \times \lambda ( x ^ { \prime \prime } )$, where $m ( x ^ { \prime } )$ is the Lebesgue measure on $\mathbf{R} ^ { k }$ and $\lambda ( x ^ { \prime \prime } )$ is a suitable reconstructing measure on the $x ^ { \prime \prime }$-plane. Then (a3) can be rewritten as an inversion formula for the $k$-plane transform, which assigns to $f$ a collection of integrals of $f$ over all $k$-dimensional planes in ${\bf R} ^ { n }$. The idea of such an application of the Calderón reproducing formula is due to M. Holschneider [a8], Sec. 12, who considered the case $n = 2$. Another choice of $\mu$ in (a3) leads to an explicit inversion of windowed X-ray transforms (cf. also X-ray transform), defined by

\begin{equation} \tag{a4} ( X _ { \nu } f ) ( x , y ) = \int _ { - \infty } ^ { \infty } f ( x + t y ) d \nu ( t ), \end{equation}

\begin{equation*} x , y \in \mathbf{R} ^ { n } \end{equation*}

with a finite Borel measure $\nu$. The transform (a4) with $d \nu ( t ) = g ( t ) d t$, where $g$ is a "nice" compactly supported function (a window function), was introduced by G.A. Kaizer and R.F. Streater [a9] in connection with applications in physics.

The Calderón reproducing formula admits various discrete versions, which serve as natural analogues of atomic decompositions and play an important role in the study of function spaces [a10].

References

[a1] A.P. Calderón, "Intermediate spaces and interpolation, the complex method" Studia Math. , 24 (1964) pp. 113–190
[a2] Y. Meyer, "Wavelets and operators" , Cambridge Univ. Press (1992)
[a3] B. Rubin, "The Calderón reproducing formula, windowed X-ray transforms and Radon transforms in $L ^ { p }$-spaces" J. Fourier Anal. Appl. , 4 (1998) pp. 175–197
[a4] D. Ryabogin, B. Rubin, "Singular integral operators generated by wavelet transforms" Integral Eq. Operator Th. (in press},)
[a5] G.B. Folland, E.M. Stein, "Hardy spaces on homogeneous groups" , Princeton Univ. Press (1982)
[a6] B. Rubin, "Fractional integrals and potentials" , Addison-Wesley (1996)
[a7] B. Rubin, "Fractional calculus and wavelet transforms in integral geometry" Fractional Calculus and Applied Analysis , 2 (1998) pp. 193–219
[a8] M. Holschneider, "Wavelets: an analysis tool" , Clarendon Press (1995)
[a9] G.A. Kaiser, R.F. Streater, "Windowed Radon transforms, analytic signals, and wave equation" C.K. Chui (ed.) , Wavelets: A Tutorial in Theory and Applications , Acad. Press (1992) pp. 399–441
[a10] M. Frazier, B. Jawerth, G. Weiss, "Littlewood–Paley theory and the study of function spaces" , CBMS Reg. Conf. Ser. Math. , 79 , Amer. Math. Soc. (1991)
How to Cite This Entry:
Calderón-type reproducing formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Calder%C3%B3n-type_reproducing_formula&oldid=23208
This article was adapted from an original article by B. Rubin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article