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Polynomials (cf. also [[Polynomial|Polynomial]]) constructed by F. Zernike [[#References|[a5]]] and by Zernike and H. Brinkman [[#References|[a6]]] for the purpose of approximating certain functions, such as the aberration function of geometrical optics, on the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z1300801.png" />. The underlying premise is that errors in circular optical elements can be quantified by mean-square deviation per unit area. Given a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z1300802.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z1300803.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z1300804.png" />, the problem of finding a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z1300805.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z1300806.png" /> which minimizes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z1300807.png" />-norm
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<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/z/z130/z130080/z1300808.png" /></td> </tr></table>
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Out of 61 formulas, 57 were replaced by TEX code.-->
  
is solved by means of orthogonal polynomials (cf. [[Orthogonal polynomials|Orthogonal polynomials]]). This means that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z1300809.png" /> there is an orthogonal basis for the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008010.png" /> of polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008011.png" />, which are orthogonal to each polynomial of lower degree (orthogonality is with respect to the inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008012.png" />). The dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008013.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008014.png" />. In the case of the disc there are at least two useful approaches to constructing orthogonal polynomials, based on the Cartesian or on the polar coordinate system. The Zernike polynomials are associated with the polar coordinate system (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008016.png" />) and with complex coordinates (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008018.png" />). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008020.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008021.png" />, the Zernike circle polynomial is
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Polynomials (cf. also [[Polynomial|Polynomial]]) constructed by F. Zernike [[#References|[a5]]] and by Zernike and H. Brinkman [[#References|[a6]]] for the purpose of approximating certain functions, such as the aberration function of geometrical optics, on the disc $D = \{ ( x , y ) \in \mathbf{R} ^ { 2 } : x ^ { 2 } + y ^ { 2 } \leq 1 \}$. The underlying premise is that errors in circular optical elements can be quantified by mean-square deviation per unit area. Given a function $f$ on $D$ and $n \in \mathbf{N} _ { 0 } = \{ 0,1,2 , \dots \}$, the problem of finding a polynomial $p ( x , y )$ of degree $n$ which minimizes the $L^{2}$-norm
  
<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/z/z130/z130080/z13008022.png" /></td> </tr></table>
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\begin{equation*} \| f - p \| _ { 2 } = \left( \int \int _ { D } | f ( x , y ) - p ( x , y ) | ^ { 2 } d x d y \right) ^ { 1 / 2 } \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008023.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008025.png" />, of the same parity as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008026.png" />. This family has been generalized to  "disc polynomials" , associated with the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008027.png" /> with arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008028.png" /> (see [[#References|[a3]]]). The formulas will be stated for the general case since they are no more complicated than for the Zernike polynomials (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008029.png" />). A convenient indexing is obtained from setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008031.png" /> for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008032.png" />. Then (using the Pochhammer symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008033.png" /> and the [[Hypergeometric function|hypergeometric function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008034.png" />), define
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is solved by means of orthogonal polynomials (cf. [[Orthogonal polynomials|Orthogonal polynomials]]). This means that for each $n \in \mathbf{N} _ { 0 }$ there is an orthogonal basis for the space $\mathcal{V} _ { n }$ of polynomials of degree $n$, which are orthogonal to each polynomial of lower degree (orthogonality is with respect to the inner product $\langle f , g \rangle = \int \int _ { D } f ( x , y ) \overline { g ( x , y ) } d x d y$). The dimension of $\mathcal{V} _ { n }$ is $n + 1$. In the case of the disc there are at least two useful approaches to constructing orthogonal polynomials, based on the Cartesian or on the polar coordinate system. The Zernike polynomials are associated with the polar coordinate system ($x = r \operatorname { cos } \theta$, $y = r \operatorname { sin } \theta$) and with complex coordinates ($z = x + i y = r e ^ { i \theta }$, $r ^ { 2 } = z \bar{z}$). For $n \in \mathbf{N} _ { 0 }$ and $m = n - 2 j$ with $j = 0 , \dots , n$, the Zernike circle polynomial is
  
<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/z/z130/z130080/z13008035.png" /></td> </tr></table>
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\begin{equation*} V _ { n } ^ { m } ( x , y ) = e ^ { i m \theta } R _ { n } ^ { m } ( r ), \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/z/z130/z130080/z13008036.png" /></td> </tr></table>
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where $R _ { n } ^ { m } ( r )$ is a polynomial of degree $n$ in $r$, of the same parity as $n$. This family has been generalized to  "disc polynomials" , associated with the weight function $( 1 - x ^ { 2 } - y ^ { 2 } ) ^ { \alpha } d x d y$ with arbitrary $\alpha &gt; - 1$ (see [[#References|[a3]]]). The formulas will be stated for the general case since they are no more complicated than for the Zernike polynomials ($\alpha = 0$). A convenient indexing is obtained from setting $n = k + l$, $m = k - l$ for arbitrary $k , l \in {\bf N} _ { 0 }$. Then (using the Pochhammer symbol $(a)_ { n } = \prod _ { i = 1 } ^ { n } ( a + i - 1 )$ and the [[Hypergeometric function|hypergeometric function]] $F$), define
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\begin{equation*} V _ { k + l } ^ { k - l } ( x , y ; \alpha ) = e ^ { i ( k - l ) \theta } R _ { k + l } ^ { k - l } ( r ; \alpha ) = \end{equation*}
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\begin{equation*} = \frac { ( \alpha + 1 ) _ { k + l } } { ( \alpha + 1 ) _ { k } ( \alpha + 1 ) _ { l } } z ^ { k } z ^ { l } F \left( - k , - l ; - k - l - \alpha ; \frac { 1 } { z \overline{z} } \right). \end{equation*}
  
 
The Zernike radial polynomial is
 
The Zernike radial polynomial is
  
<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/z/z130/z130080/z13008037.png" /></td> </tr></table>
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\begin{equation*} R _ { k + l } ^ { k - l } ( r ; \alpha ) = \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/z/z130/z130080/z13008038.png" /></td> </tr></table>
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\begin{equation*} = \frac { ( \alpha + 1 ) _ { k + l } } { ( \alpha + 1 ) _ { k } ( \alpha + 1 ) _ { l } } \sum _ { j = 0 } ^ { \operatorname { min } ( k , l ) } \frac { ( - k ) _ { j } ( - l ) _j} { ( - k - l - \alpha )_j j ! } r ^ { k + l - 2 j }. \end{equation*}
  
The normalization of the polynomials comes from the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008039.png" />. The orthogonality relations are
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The normalization of the polynomials comes from the equation $V _ { k + l } ^ { k - l } ( 1,0 ; \alpha ) = 1$. The orthogonality relations are
  
<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/z/z130/z130080/z13008040.png" /></td> </tr></table>
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<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/z/z130/z130080/z13008040.png"/></td> </tr></table>
  
<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/z/z130/z130080/z13008041.png" /></td> </tr></table>
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<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/z/z130/z130080/z13008041.png"/></td> </tr></table>
  
The polynomials can be expressed in terms of [[Jacobi polynomials|Jacobi polynomials]]: for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008042.png" />,
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The polynomials can be expressed in terms of [[Jacobi polynomials|Jacobi polynomials]]: for $k \geq l $,
  
<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/z/z130/z130080/z13008043.png" /></td> </tr></table>
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\begin{equation*} R _ { k + l } ^ { k - l } ( r ; \alpha ) = \frac { l ! } { ( \alpha + 1 ) _ { l } } r ^ { k - l } P _ { l } ^ { ( \alpha , k - l ) } ( 2 r ^ { 2 } - 1 ), \end{equation*}
  
 
and satisfy a Rodrigues formula:
 
and satisfy a Rodrigues formula:
  
<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/z/z130/z130080/z13008044.png" /></td> </tr></table>
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\begin{equation*} V _ { k + l } ^ { k - l } ( x , y ; \alpha ) = \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/z/z130/z130080/z13008045.png" /></td> </tr></table>
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\begin{equation*} = \frac { ( - 1 ) ^ { k + l } } { ( \alpha + 1 ) _ { k + l } } ( 1 - z \overline{z} ) ^ { - \alpha } ( \frac { \partial } { \partial z } ) ^ { l } ( \frac { \partial } { \partial \overline{z} } ) ^ { k } ( 1 - z \overline{z} ) ^ { k + l + \alpha }. \end{equation*}
  
The orthogonal polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008046.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008047.png" />) satisfy a differential equation:
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The orthogonal polynomials of degree $n$ (that is, $V _ { n } = \operatorname { span } \left\{ V _ { n } ^ { n - 2 j } : 0 \leq j \leq n \right\}$) satisfy a differential 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/z/z130/z130080/z13008048.png" /></td> </tr></table>
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\begin{equation*} \left( 4 \frac { \partial ^ { 2 } } { \partial z \partial \overline{z} } - \mathcal{D} ^ { 2 } - 2 ( \alpha + 1 ) \mathcal{D} \right) 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/z/z130/z130080/z13008049.png" /></td> </tr></table>
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\begin{equation*} = - n ( n + 2 + 2 \alpha ) f , \mathcal{D} = z \frac { \partial } { \partial z } + \bar{z} \frac { \partial } { \partial \bar{z} }. \end{equation*}
  
There is an important [[Integral transform|integral transform]] used in the diffraction theory of aberrations (see [[#References|[a1]]], Chap. 9): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008051.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008052.png" />, then
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There is an important [[Integral transform|integral transform]] used in the diffraction theory of aberrations (see [[#References|[a1]]], Chap. 9): Let $k , l \in {\bf N} _ { 0 }$, $k \geq l $, and $s &gt; 0$, 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/z/z130/z130080/z13008053.png" /></td> </tr></table>
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\begin{equation*} \int _ { 0 } ^ { 1 } R _ { k + l } ^ { k - l } ( r ; \alpha ) J _ { k - l } ( r s ) ( 1 - r ^ { 2 } ) ^ { \alpha } r d r = \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/z/z130/z130080/z13008054.png" /></td> </tr></table>
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\begin{equation*} = \frac { ( - 1 ) ^ { l } } { 2 } \Gamma ( \alpha + 1 ) \left( \frac { 2 } { s } \right) ^ { \alpha + 1 } J _ { k + l + \alpha + 1 } ( s ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008055.png" /> denotes the Bessel function of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008056.png" /> (cf. [[Bessel functions|Bessel functions]]). The coefficients of the orthogonal expansion of an aberration function in terms of the Zernike polynomials are related to the so-called primary aberrations (such as astigmatism, coma, distortion), see [[#References|[a1]]], Chap. 5. The disc polynomials for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008057.png" /> appear as [[Spherical functions|spherical functions]] on the homogeneous spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008058.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130080/z13008059.png" /> denotes the unitary group, see [[#References|[a2]]], Vol. 2, Sec. 11.5, pp. 359–363). The Zernike polynomials are key tools in two-dimensional [[Tomography|tomography]]; see [[#References|[a4]]].
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where $J _ { a }$ denotes the Bessel function of index $a$ (cf. [[Bessel functions|Bessel functions]]). The coefficients of the orthogonal expansion of an aberration function in terms of the Zernike polynomials are related to the so-called primary aberrations (such as astigmatism, coma, distortion), see [[#References|[a1]]], Chap. 5. The disc polynomials for $\alpha \in \mathbf{N} _ { 0 }$ appear as [[Spherical functions|spherical functions]] on the homogeneous spaces $U ( \alpha + 2 ) / U ( \alpha + 1 )$ (where $U$ denotes the unitary group, see [[#References|[a2]]], Vol. 2, Sec. 11.5, pp. 359–363). The Zernike polynomials are key tools in two-dimensional [[Tomography|tomography]]; see [[#References|[a4]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Born,  E. Wolf,  "Principles of optics" , Pergamon  (1965)  (Edition: Third)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Klimyk,  N. Vilenkin,  "Representations of Lie groups and special functions" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Koornwinder,  "Two-variable analogues of the classical orthogonal polynomials"  R. Askey (ed.) , ''Theory and Applications of Special Functions'' , Acad. Press  (1975)  pp. 435–495</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Marr,  "On the reconstruction of a function on a circular domain from a sampling of its line integrals"  ''J. Math. Anal. Appl.'' , '''45'''  (1974)  pp. 357–374</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  F. Zernike,  "Beugungstheorie des Schneidensverfahrens und seiner verbesserten Form, der Phasenkontrastmethode"  ''Physica'' , '''1'''  (1934)  pp. 689–704</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  F. Zernike,  H. Brinkman,  "Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome"  ''Proc. K. Akad. Wetensch.'' , '''38'''  (1935)  pp. 161–170</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Born,  E. Wolf,  "Principles of optics" , Pergamon  (1965)  (Edition: Third)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Klimyk,  N. Vilenkin,  "Representations of Lie groups and special functions" , Kluwer Acad. Publ.  (1993)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  T. Koornwinder,  "Two-variable analogues of the classical orthogonal polynomials"  R. Askey (ed.) , ''Theory and Applications of Special Functions'' , Acad. Press  (1975)  pp. 435–495</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  R. Marr,  "On the reconstruction of a function on a circular domain from a sampling of its line integrals"  ''J. Math. Anal. Appl.'' , '''45'''  (1974)  pp. 357–374</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  F. Zernike,  "Beugungstheorie des Schneidensverfahrens und seiner verbesserten Form, der Phasenkontrastmethode"  ''Physica'' , '''1'''  (1934)  pp. 689–704</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  F. Zernike,  H. Brinkman,  "Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome"  ''Proc. K. Akad. Wetensch.'' , '''38'''  (1935)  pp. 161–170</td></tr></table>

Revision as of 15:19, 1 July 2020

Polynomials (cf. also Polynomial) constructed by F. Zernike [a5] and by Zernike and H. Brinkman [a6] for the purpose of approximating certain functions, such as the aberration function of geometrical optics, on the disc $D = \{ ( x , y ) \in \mathbf{R} ^ { 2 } : x ^ { 2 } + y ^ { 2 } \leq 1 \}$. The underlying premise is that errors in circular optical elements can be quantified by mean-square deviation per unit area. Given a function $f$ on $D$ and $n \in \mathbf{N} _ { 0 } = \{ 0,1,2 , \dots \}$, the problem of finding a polynomial $p ( x , y )$ of degree $n$ which minimizes the $L^{2}$-norm

\begin{equation*} \| f - p \| _ { 2 } = \left( \int \int _ { D } | f ( x , y ) - p ( x , y ) | ^ { 2 } d x d y \right) ^ { 1 / 2 } \end{equation*}

is solved by means of orthogonal polynomials (cf. Orthogonal polynomials). This means that for each $n \in \mathbf{N} _ { 0 }$ there is an orthogonal basis for the space $\mathcal{V} _ { n }$ of polynomials of degree $n$, which are orthogonal to each polynomial of lower degree (orthogonality is with respect to the inner product $\langle f , g \rangle = \int \int _ { D } f ( x , y ) \overline { g ( x , y ) } d x d y$). The dimension of $\mathcal{V} _ { n }$ is $n + 1$. In the case of the disc there are at least two useful approaches to constructing orthogonal polynomials, based on the Cartesian or on the polar coordinate system. The Zernike polynomials are associated with the polar coordinate system ($x = r \operatorname { cos } \theta$, $y = r \operatorname { sin } \theta$) and with complex coordinates ($z = x + i y = r e ^ { i \theta }$, $r ^ { 2 } = z \bar{z}$). For $n \in \mathbf{N} _ { 0 }$ and $m = n - 2 j$ with $j = 0 , \dots , n$, the Zernike circle polynomial is

\begin{equation*} V _ { n } ^ { m } ( x , y ) = e ^ { i m \theta } R _ { n } ^ { m } ( r ), \end{equation*}

where $R _ { n } ^ { m } ( r )$ is a polynomial of degree $n$ in $r$, of the same parity as $n$. This family has been generalized to "disc polynomials" , associated with the weight function $( 1 - x ^ { 2 } - y ^ { 2 } ) ^ { \alpha } d x d y$ with arbitrary $\alpha > - 1$ (see [a3]). The formulas will be stated for the general case since they are no more complicated than for the Zernike polynomials ($\alpha = 0$). A convenient indexing is obtained from setting $n = k + l$, $m = k - l$ for arbitrary $k , l \in {\bf N} _ { 0 }$. Then (using the Pochhammer symbol $(a)_ { n } = \prod _ { i = 1 } ^ { n } ( a + i - 1 )$ and the hypergeometric function $F$), define

\begin{equation*} V _ { k + l } ^ { k - l } ( x , y ; \alpha ) = e ^ { i ( k - l ) \theta } R _ { k + l } ^ { k - l } ( r ; \alpha ) = \end{equation*}

\begin{equation*} = \frac { ( \alpha + 1 ) _ { k + l } } { ( \alpha + 1 ) _ { k } ( \alpha + 1 ) _ { l } } z ^ { k } z ^ { l } F \left( - k , - l ; - k - l - \alpha ; \frac { 1 } { z \overline{z} } \right). \end{equation*}

The Zernike radial polynomial is

\begin{equation*} R _ { k + l } ^ { k - l } ( r ; \alpha ) = \end{equation*}

\begin{equation*} = \frac { ( \alpha + 1 ) _ { k + l } } { ( \alpha + 1 ) _ { k } ( \alpha + 1 ) _ { l } } \sum _ { j = 0 } ^ { \operatorname { min } ( k , l ) } \frac { ( - k ) _ { j } ( - l ) _j} { ( - k - l - \alpha )_j j ! } r ^ { k + l - 2 j }. \end{equation*}

The normalization of the polynomials comes from the equation $V _ { k + l } ^ { k - l } ( 1,0 ; \alpha ) = 1$. The orthogonality relations are

The polynomials can be expressed in terms of Jacobi polynomials: for $k \geq l $,

\begin{equation*} R _ { k + l } ^ { k - l } ( r ; \alpha ) = \frac { l ! } { ( \alpha + 1 ) _ { l } } r ^ { k - l } P _ { l } ^ { ( \alpha , k - l ) } ( 2 r ^ { 2 } - 1 ), \end{equation*}

and satisfy a Rodrigues formula:

\begin{equation*} V _ { k + l } ^ { k - l } ( x , y ; \alpha ) = \end{equation*}

\begin{equation*} = \frac { ( - 1 ) ^ { k + l } } { ( \alpha + 1 ) _ { k + l } } ( 1 - z \overline{z} ) ^ { - \alpha } ( \frac { \partial } { \partial z } ) ^ { l } ( \frac { \partial } { \partial \overline{z} } ) ^ { k } ( 1 - z \overline{z} ) ^ { k + l + \alpha }. \end{equation*}

The orthogonal polynomials of degree $n$ (that is, $V _ { n } = \operatorname { span } \left\{ V _ { n } ^ { n - 2 j } : 0 \leq j \leq n \right\}$) satisfy a differential equation:

\begin{equation*} \left( 4 \frac { \partial ^ { 2 } } { \partial z \partial \overline{z} } - \mathcal{D} ^ { 2 } - 2 ( \alpha + 1 ) \mathcal{D} \right) f = \end{equation*}

\begin{equation*} = - n ( n + 2 + 2 \alpha ) f , \mathcal{D} = z \frac { \partial } { \partial z } + \bar{z} \frac { \partial } { \partial \bar{z} }. \end{equation*}

There is an important integral transform used in the diffraction theory of aberrations (see [a1], Chap. 9): Let $k , l \in {\bf N} _ { 0 }$, $k \geq l $, and $s > 0$, then

\begin{equation*} \int _ { 0 } ^ { 1 } R _ { k + l } ^ { k - l } ( r ; \alpha ) J _ { k - l } ( r s ) ( 1 - r ^ { 2 } ) ^ { \alpha } r d r = \end{equation*}

\begin{equation*} = \frac { ( - 1 ) ^ { l } } { 2 } \Gamma ( \alpha + 1 ) \left( \frac { 2 } { s } \right) ^ { \alpha + 1 } J _ { k + l + \alpha + 1 } ( s ), \end{equation*}

where $J _ { a }$ denotes the Bessel function of index $a$ (cf. Bessel functions). The coefficients of the orthogonal expansion of an aberration function in terms of the Zernike polynomials are related to the so-called primary aberrations (such as astigmatism, coma, distortion), see [a1], Chap. 5. The disc polynomials for $\alpha \in \mathbf{N} _ { 0 }$ appear as spherical functions on the homogeneous spaces $U ( \alpha + 2 ) / U ( \alpha + 1 )$ (where $U$ denotes the unitary group, see [a2], Vol. 2, Sec. 11.5, pp. 359–363). The Zernike polynomials are key tools in two-dimensional tomography; see [a4].

References

[a1] M. Born, E. Wolf, "Principles of optics" , Pergamon (1965) (Edition: Third)
[a2] A. Klimyk, N. Vilenkin, "Representations of Lie groups and special functions" , Kluwer Acad. Publ. (1993)
[a3] T. Koornwinder, "Two-variable analogues of the classical orthogonal polynomials" R. Askey (ed.) , Theory and Applications of Special Functions , Acad. Press (1975) pp. 435–495
[a4] R. Marr, "On the reconstruction of a function on a circular domain from a sampling of its line integrals" J. Math. Anal. Appl. , 45 (1974) pp. 357–374
[a5] F. Zernike, "Beugungstheorie des Schneidensverfahrens und seiner verbesserten Form, der Phasenkontrastmethode" Physica , 1 (1934) pp. 689–704
[a6] F. Zernike, H. Brinkman, "Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome" Proc. K. Akad. Wetensch. , 38 (1935) pp. 161–170
How to Cite This Entry:
Zernike polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zernike_polynomials&oldid=17503
This article was adapted from an original article by Charles F. Dunkl (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article