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The general name for polynomials orthogonal on the circle, over a contour or over an area. Unlike the case of orthogonality in a real domain, the polynomials of the three kinds of systems mentioned can have imaginary coefficients and are examined for all complex values of the independent variable. A characteristic feature of cases of orthogonality on a complex domain is that analytic functions of a complex variable which satisfy certain supplementary conditions in a neighbourhood of the boundary of the domain of analyticity can usually be expanded in a Fourier series in these systems (cf. [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]]).
 
The general name for polynomials orthogonal on the circle, over a contour or over an area. Unlike the case of orthogonality in a real domain, the polynomials of the three kinds of systems mentioned can have imaginary coefficients and are examined for all complex values of the independent variable. A characteristic feature of cases of orthogonality on a complex domain is that analytic functions of a complex variable which satisfy certain supplementary conditions in a neighbourhood of the boundary of the domain of analyticity can usually be expanded in a Fourier series in these systems (cf. [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]]).
  
 
==Orthogonal polynomials on the circle.==
 
==Orthogonal polynomials on the circle.==
A system of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o0703501.png" /> having positive leading coefficient and satisfying the orthogonality (usually orthonormality) condition:
+
A system of polynomials $  \{ \phi _ {n} \} $
 +
having positive leading coefficient and satisfying the orthogonality (usually orthonormality) condition:
 +
 
 +
$$
  
<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/o/o070/o070350/o0703502.png" /></td> </tr></table>
+
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi } \phi _ {n} ( e ^ {i \theta } )
 +
{\phi _ {m} ( e ^ {i \theta } ) } bar  d \mu ( \theta )  = \delta _ {nm} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o0703503.png" /> is a bounded non-decreasing function on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o0703504.png" /> with an infinite number of points of growth, called a distribution function, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o0703505.png" /> is the Kronecker symbol. A recurrence relation and the analogue of the [[Christoffel–Darboux formula|Christoffel–Darboux formula]] holds for the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o0703506.png" />, in the same way as in the case of orthogonality on an interval.
+
where $  \mu $
 +
is a bounded non-decreasing function on the interval $  [ 0, 2 \pi ] $
 +
with an infinite number of points of growth, called a distribution function, while $  \delta _ {nm} $
 +
is the Kronecker symbol. A recurrence relation and the analogue of the [[Christoffel–Darboux formula|Christoffel–Darboux formula]] holds for the polynomials $  \{ \phi _ {n} \} $,  
 +
in the same way as in the case of orthogonality on an interval.
  
 
Asymptotic properties are examined under the condition
 
Asymptotic properties are examined under the condition
  
<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/o/o070/o070350/o0703507.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^ { {2 }  \pi }  \mathop{\rm ln}  \mu  ^  \prime  ( \theta )  d \theta  > - \infty .
 +
$$
  
 
The case of orthogonality on the circle as a periodic case has been studied in sufficient detail, and the results of the approximation of periodic functions by trigonometric polynomials have been successfully used.
 
The case of orthogonality on the circle as a periodic case has been studied in sufficient detail, and the results of the approximation of periodic functions by trigonometric polynomials have been successfully used.
  
Let the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o0703508.png" /> be orthonormal on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o0703509.png" /> with differential weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035010.png" />, and let the weight function on the circle have the form
+
Let the polynomials $  \{ P _ {n} \} $
 +
be orthonormal on the segment $  [- 1, 1] $
 +
with differential weight function $  h $,  
 +
and let the weight function on the circle have 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/o/o070/o070350/o07035011.png" /></td> </tr></table>
+
$$
 +
\mu  ^  \prime  ( \theta )  = h( \cos  \theta )  | \sin  \theta |.
 +
$$
  
Under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035012.png" />, the Szegö formula
+
Under the condition $  x = ( z  ^ {2} + 1)/2z $,  
 +
the Szegö 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/o/o070/o070350/o07035013.png" /></td> </tr></table>
+
$$
 +
P _ {n} ( x)  =
 +
\frac{1}{\sqrt {2 \pi } }
 +
\left [ 1 +
 +
\frac{\phi _ {2n} ( 0) }{\alpha _ {2n} }
 +
\right ]  ^ {-} 1/2 \left [
 +
\frac{1}{z  ^ {n} }
 +
\phi _ {2n} ( z) + z  ^ {n}
 +
\phi _ {2n} \left (
 +
\frac{1}{z}
 +
\right )  \right ]
 +
$$
  
holds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035014.png" /> is the leading coefficient of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035015.png" />.
+
holds, where $  \alpha _ {2n} $
 +
is the leading coefficient of the polynomial $  \phi _ {2n} $.
  
If an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035016.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035017.png" /> has non-tangential boundary values on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035018.png" />, then under certain supplementary conditions the expansion
+
If an analytic function $  f $
 +
in the disc $  | z | < 1 $
 +
has non-tangential boundary values on the circle $  | z | = 1 $,  
 +
then under certain supplementary conditions the expansion
  
<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/o/o070/o070350/o07035019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
f( z)  = \sum _ { n= } 0 ^  \infty  a _ {n} \phi _ {n} ( z),\ \
 +
| z | < 1 ,
 +
$$
  
 
holds; its coefficients are defined by the formula
 
holds; its coefficients are defined by the 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/o/o070/o070350/o07035020.png" /></td> </tr></table>
+
$$
 +
a _ {n}  =
 +
\frac{1}{2 \pi }
 +
\int\limits _ { 0 } ^ { {2 }  \pi } f( e ^ {i \theta } ) {\phi _ {n} ( e ^ {i \theta } ) } bar  d \mu ( \theta ).
 +
$$
  
Series of the form (1) are direct generalizations of Taylor series: if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035022.png" />. Given certain conditions on the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035023.png" />, the series (1) converges or diverges simultaneously with the Taylor series of the same function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035024.png" /> at the points of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035025.png" />, i.e. the theorem on equiconvergence of these two series holds.
+
Series of the form (1) are direct generalizations of Taylor series: if $  \mu ( \theta ) = \theta $,  
 +
$  \phi _ {n} ( z) \equiv z  ^ {n} $.  
 +
Given certain conditions on the distribution function $  \mu $,  
 +
the series (1) converges or diverges simultaneously with the Taylor series of the same function $  f $
 +
at the points of the circle $  | z | = 1 $,  
 +
i.e. the theorem on equiconvergence of these two series holds.
  
 
==Orthogonal polynomials over a contour.==
 
==Orthogonal polynomials over a contour.==
A system of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035026.png" /> having positive leading coefficient and satisfying the condition
+
A system of polynomials $  \{ P _ {n} \} $
 +
having positive leading coefficient and satisfying the condition
 +
 
 +
$$
  
<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/o/o070/o070350/o07035027.png" /></td> </tr></table>
+
\frac{1}{2 \pi }
 +
\int\limits _  \Gamma  P _ {n} ( z) \overline{ {P _ {m} ( z) }}\; h( z) \
 +
| dz |  = \delta _ {nm} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035028.png" /> is a rectifiable Jordan curve (usually closed) in the complex plane, while the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035029.png" /> is Lebesgue integrable and positive almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035030.png" />.
+
where $  \Gamma $
 +
is a rectifiable Jordan curve (usually closed) in the complex plane, while the weight function $  h $
 +
is Lebesgue integrable and positive almost-everywhere on $  \Gamma $.
  
Let, in the simply-connected bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035031.png" /> bounded by the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035032.png" />, an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035033.png" /> be given whose boundary values on the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035034.png" /> are square integrable with respect to the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035035.png" />. Using the formula for the coefficients,
+
Let, in the simply-connected bounded domain $  G $
 +
bounded by the curve $  \Gamma $,  
 +
an analytic function $  f $
 +
be given whose boundary values on the contour $  \Gamma $
 +
are square integrable with respect to the weight function $  h $.  
 +
Using the formula for the coefficients,
  
<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/o/o070/o070350/o07035036.png" /></td> </tr></table>
+
$$
 +
a _ {n}  =
 +
\frac{1}{2 \pi }
 +
\int\limits _  \Gamma  f( \zeta ) \overline{ {P _ {n} ( \zeta ) }}\; h(
 +
\zeta )  | d \zeta | ,
 +
$$
  
 
a Fourier series in the orthogonal polynomials,
 
a Fourier series in the orthogonal polynomials,
  
<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/o/o070/o070350/o07035037.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\sum _ { n= } 0 ^  \infty  a _ {n} P _ {n} ,
 +
$$
  
 
then corresponds to this function. These series are a natural generalization of Taylor series with respect to the orthogonality property in the case of a simply-connected domain, and serve as a representation of analytic functions. If the completeness condition
 
then corresponds to this function. These series are a natural generalization of Taylor series with respect to the orthogonality property in the case of a simply-connected domain, and serve as a representation of analytic functions. If the completeness condition
  
<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/o/o070/o070350/o07035038.png" /></td> </tr></table>
+
$$
 +
\inf _ {\{ Q _ {n} \} } \int\limits _  \Gamma  h( z) \
 +
| f( z) - Q _ {n} ( z)  |  ^ {2} |  dz |  = 0
 +
$$
  
is fulfilled, where the infimum is taken over the set of all polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035039.png" />, then the series (2) converges in the mean to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035040.png" /> along the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035041.png" /> with weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035042.png" /> and, under certain supplementary conditions, inside the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035043.png" /> as well.
+
is fulfilled, where the infimum is taken over the set of all polynomials $  Q _ {n} $,  
 +
then the series (2) converges in the mean to the function $  f $
 +
along the contour $  \Gamma $
 +
with weight $  h $
 +
and, under certain supplementary conditions, inside the domain $  G $
 +
as well.
  
 
==Orthogonal polynomials over a domain.==
 
==Orthogonal polynomials over a domain.==
A system of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035044.png" /> having positive leading coefficient and satisfying the condition
+
A system of polynomials $  \{ K _ {n} \} $
 +
having positive leading coefficient and satisfying the condition
  
<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/o/o070/o070350/o07035045.png" /></td> </tr></table>
+
$$
 +
{\int\limits \int\limits } _ { G } K _ {n} ( z) \overline{ {K _ {m} ( z) }}\; h( z)  dx  dy  = \delta _ {nm} ,
 +
$$
  
where the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035046.png" /> is non-negative, integrable with respect to the area of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035047.png" />, and not equal to zero. If the completeness condition
+
where the weight function $  h $
 +
is non-negative, integrable with respect to the area of a bounded domain $  G $,  
 +
and not equal to zero. If the completeness condition
  
<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/o/o070/o070350/o07035048.png" /></td> </tr></table>
+
$$
 +
\inf _ {\{ Q _ {n} \} } {\int\limits \int\limits } _ { G } h( z) \
 +
| f( z) - Q _ {n} ( z) |  ^ {2}  dx  dy  = 0
 +
$$
  
is fulfilled, where the infimum is taken over the set of all polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035049.png" />, then the Fourier series in the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035050.png" /> of an analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035051.png" /> in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035052.png" /> converges in the mean (with respect to the area of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035053.png" />) with weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035054.png" /> to this function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035055.png" /> and, under certain supplementary conditions, inside the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035056.png" /> as well.
+
is fulfilled, where the infimum is taken over the set of all polynomials $  Q _ {n} $,  
 +
then the Fourier series in the polynomials $  \{ K _ {n} \} $
 +
of an analytic function $  f $
 +
in a simply-connected domain $  G $
 +
converges in the mean (with respect to the area of the domain $  G $)  
 +
with weight $  h $
 +
to this function $  f $
 +
and, under certain supplementary conditions, inside the domain $  G $
 +
as well.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  G. Szegö,  "Beiträge zur Theorie der Toeplitzschen Formen, I"  ''Math. Z.'' , '''6'''  (1920)  pp. 167–202  (Also: Collected Works, Vol. 1, Birkhäuser, 1982, pp. 237–272)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  G. Szegö,  "Beiträge zur Theorie der Toeplitzschen Formen, II"  ''Math. Z.'' , '''9'''  (1921)  pp. 167–190  (Also: Collected Works, Vol. 1, Birkhäuser, 1982, pp. 279–305)</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  G. Szegö,  "Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehören"  ''Math. Z.'' , '''9'''  (1921)  pp. 218–270  (Also: Collected Works, Vol. 1, Birkhäuser, 1982, pp. 316–368)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Carleman,  "Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen"  ''Ark. for Mat., Astr. och Fys.'' , '''17''' :  9  (1922–1923)  pp. 1–30</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Szegö,  "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Ya.L. Geronimus,  "Polynomials orthogonal on a circle and interval" , Pergamon  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.I. Smirnov,  "On the theory of orthogonal polynomials of a complex variable"  ''Zh. Leningrad. Fiz.-Mat. Obshch.'' , '''2''' :  1  (1928)  pp. 155–179  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P.P. Korovkin,  "On polynomials orthogonal on a rectifiable contour in the presence of a weight"  ''Mat. Sb.'' , '''9''' :  3  (1941)  pp. 469–485  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.K. Suetin,  "Fundamental properties of polynomials orthogonal on a contour"  ''Russian Math.Surveys'' , '''21''' :  2  (1966)  pp. 35–83  ''Uspekhi Mat. Nauk'' , '''21''' :  2  (1966)  pp. 41–88</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  P.K. Suetin,  "Polynomials orthogonal over a region and Bieberbach polynomials"  ''Proc. Steklov Inst. Math.'' , '''100'''  (1974)  ''Trudy Mat. Inst. Steklov.'' , '''100'''  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  G. Szegö,  "Beiträge zur Theorie der Toeplitzschen Formen, I"  ''Math. Z.'' , '''6'''  (1920)  pp. 167–202  (Also: Collected Works, Vol. 1, Birkhäuser, 1982, pp. 237–272)</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  G. Szegö,  "Beiträge zur Theorie der Toeplitzschen Formen, II"  ''Math. Z.'' , '''9'''  (1921)  pp. 167–190  (Also: Collected Works, Vol. 1, Birkhäuser, 1982, pp. 279–305)</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top">  G. Szegö,  "Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehören"  ''Math. Z.'' , '''9'''  (1921)  pp. 218–270  (Also: Collected Works, Vol. 1, Birkhäuser, 1982, pp. 316–368)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Carleman,  "Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen"  ''Ark. for Mat., Astr. och Fys.'' , '''17''' :  9  (1922–1923)  pp. 1–30</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Szegö,  "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Ya.L. Geronimus,  "Polynomials orthogonal on a circle and interval" , Pergamon  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.I. Smirnov,  "On the theory of orthogonal polynomials of a complex variable"  ''Zh. Leningrad. Fiz.-Mat. Obshch.'' , '''2''' :  1  (1928)  pp. 155–179  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P.P. Korovkin,  "On polynomials orthogonal on a rectifiable contour in the presence of a weight"  ''Mat. Sb.'' , '''9''' :  3  (1941)  pp. 469–485  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  P.K. Suetin,  "Fundamental properties of polynomials orthogonal on a contour"  ''Russian Math.Surveys'' , '''21''' :  2  (1966)  pp. 35–83  ''Uspekhi Mat. Nauk'' , '''21''' :  2  (1966)  pp. 41–88</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  P.K. Suetin,  "Polynomials orthogonal over a region and Bieberbach polynomials"  ''Proc. Steklov Inst. Math.'' , '''100'''  (1974)  ''Trudy Mat. Inst. Steklov.'' , '''100'''  (1971)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:04, 6 June 2020


The general name for polynomials orthogonal on the circle, over a contour or over an area. Unlike the case of orthogonality in a real domain, the polynomials of the three kinds of systems mentioned can have imaginary coefficients and are examined for all complex values of the independent variable. A characteristic feature of cases of orthogonality on a complex domain is that analytic functions of a complex variable which satisfy certain supplementary conditions in a neighbourhood of the boundary of the domain of analyticity can usually be expanded in a Fourier series in these systems (cf. Fourier series in orthogonal polynomials).

Orthogonal polynomials on the circle.

A system of polynomials $ \{ \phi _ {n} \} $ having positive leading coefficient and satisfying the orthogonality (usually orthonormality) condition:

$$ \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } \phi _ {n} ( e ^ {i \theta } ) {\phi _ {m} ( e ^ {i \theta } ) } bar d \mu ( \theta ) = \delta _ {nm} , $$

where $ \mu $ is a bounded non-decreasing function on the interval $ [ 0, 2 \pi ] $ with an infinite number of points of growth, called a distribution function, while $ \delta _ {nm} $ is the Kronecker symbol. A recurrence relation and the analogue of the Christoffel–Darboux formula holds for the polynomials $ \{ \phi _ {n} \} $, in the same way as in the case of orthogonality on an interval.

Asymptotic properties are examined under the condition

$$ \int\limits _ { 0 } ^ { {2 } \pi } \mathop{\rm ln} \mu ^ \prime ( \theta ) d \theta > - \infty . $$

The case of orthogonality on the circle as a periodic case has been studied in sufficient detail, and the results of the approximation of periodic functions by trigonometric polynomials have been successfully used.

Let the polynomials $ \{ P _ {n} \} $ be orthonormal on the segment $ [- 1, 1] $ with differential weight function $ h $, and let the weight function on the circle have the form

$$ \mu ^ \prime ( \theta ) = h( \cos \theta ) | \sin \theta |. $$

Under the condition $ x = ( z ^ {2} + 1)/2z $, the Szegö formula

$$ P _ {n} ( x) = \frac{1}{\sqrt {2 \pi } } \left [ 1 + \frac{\phi _ {2n} ( 0) }{\alpha _ {2n} } \right ] ^ {-} 1/2 \left [ \frac{1}{z ^ {n} } \phi _ {2n} ( z) + z ^ {n} \phi _ {2n} \left ( \frac{1}{z} \right ) \right ] $$

holds, where $ \alpha _ {2n} $ is the leading coefficient of the polynomial $ \phi _ {2n} $.

If an analytic function $ f $ in the disc $ | z | < 1 $ has non-tangential boundary values on the circle $ | z | = 1 $, then under certain supplementary conditions the expansion

$$ \tag{1 } f( z) = \sum _ { n= } 0 ^ \infty a _ {n} \phi _ {n} ( z),\ \ | z | < 1 , $$

holds; its coefficients are defined by the formula

$$ a _ {n} = \frac{1}{2 \pi } \int\limits _ { 0 } ^ { {2 } \pi } f( e ^ {i \theta } ) {\phi _ {n} ( e ^ {i \theta } ) } bar d \mu ( \theta ). $$

Series of the form (1) are direct generalizations of Taylor series: if $ \mu ( \theta ) = \theta $, $ \phi _ {n} ( z) \equiv z ^ {n} $. Given certain conditions on the distribution function $ \mu $, the series (1) converges or diverges simultaneously with the Taylor series of the same function $ f $ at the points of the circle $ | z | = 1 $, i.e. the theorem on equiconvergence of these two series holds.

Orthogonal polynomials over a contour.

A system of polynomials $ \{ P _ {n} \} $ having positive leading coefficient and satisfying the condition

$$ \frac{1}{2 \pi } \int\limits _ \Gamma P _ {n} ( z) \overline{ {P _ {m} ( z) }}\; h( z) \ | dz | = \delta _ {nm} , $$

where $ \Gamma $ is a rectifiable Jordan curve (usually closed) in the complex plane, while the weight function $ h $ is Lebesgue integrable and positive almost-everywhere on $ \Gamma $.

Let, in the simply-connected bounded domain $ G $ bounded by the curve $ \Gamma $, an analytic function $ f $ be given whose boundary values on the contour $ \Gamma $ are square integrable with respect to the weight function $ h $. Using the formula for the coefficients,

$$ a _ {n} = \frac{1}{2 \pi } \int\limits _ \Gamma f( \zeta ) \overline{ {P _ {n} ( \zeta ) }}\; h( \zeta ) | d \zeta | , $$

a Fourier series in the orthogonal polynomials,

$$ \tag{2 } \sum _ { n= } 0 ^ \infty a _ {n} P _ {n} , $$

then corresponds to this function. These series are a natural generalization of Taylor series with respect to the orthogonality property in the case of a simply-connected domain, and serve as a representation of analytic functions. If the completeness condition

$$ \inf _ {\{ Q _ {n} \} } \int\limits _ \Gamma h( z) \ | f( z) - Q _ {n} ( z) | ^ {2} | dz | = 0 $$

is fulfilled, where the infimum is taken over the set of all polynomials $ Q _ {n} $, then the series (2) converges in the mean to the function $ f $ along the contour $ \Gamma $ with weight $ h $ and, under certain supplementary conditions, inside the domain $ G $ as well.

Orthogonal polynomials over a domain.

A system of polynomials $ \{ K _ {n} \} $ having positive leading coefficient and satisfying the condition

$$ {\int\limits \int\limits } _ { G } K _ {n} ( z) \overline{ {K _ {m} ( z) }}\; h( z) dx dy = \delta _ {nm} , $$

where the weight function $ h $ is non-negative, integrable with respect to the area of a bounded domain $ G $, and not equal to zero. If the completeness condition

$$ \inf _ {\{ Q _ {n} \} } {\int\limits \int\limits } _ { G } h( z) \ | f( z) - Q _ {n} ( z) | ^ {2} dx dy = 0 $$

is fulfilled, where the infimum is taken over the set of all polynomials $ Q _ {n} $, then the Fourier series in the polynomials $ \{ K _ {n} \} $ of an analytic function $ f $ in a simply-connected domain $ G $ converges in the mean (with respect to the area of the domain $ G $) with weight $ h $ to this function $ f $ and, under certain supplementary conditions, inside the domain $ G $ as well.

References

[1a] G. Szegö, "Beiträge zur Theorie der Toeplitzschen Formen, I" Math. Z. , 6 (1920) pp. 167–202 (Also: Collected Works, Vol. 1, Birkhäuser, 1982, pp. 237–272)
[1b] G. Szegö, "Beiträge zur Theorie der Toeplitzschen Formen, II" Math. Z. , 9 (1921) pp. 167–190 (Also: Collected Works, Vol. 1, Birkhäuser, 1982, pp. 279–305)
[1c] G. Szegö, "Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehören" Math. Z. , 9 (1921) pp. 218–270 (Also: Collected Works, Vol. 1, Birkhäuser, 1982, pp. 316–368)
[2] T. Carleman, "Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen" Ark. for Mat., Astr. och Fys. , 17 : 9 (1922–1923) pp. 1–30
[3] G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[4] Ya.L. Geronimus, "Polynomials orthogonal on a circle and interval" , Pergamon (1960) (Translated from Russian)
[5] V.I. Smirnov, "On the theory of orthogonal polynomials of a complex variable" Zh. Leningrad. Fiz.-Mat. Obshch. , 2 : 1 (1928) pp. 155–179 (In Russian)
[6] P.P. Korovkin, "On polynomials orthogonal on a rectifiable contour in the presence of a weight" Mat. Sb. , 9 : 3 (1941) pp. 469–485 (In Russian)
[7] P.K. Suetin, "Fundamental properties of polynomials orthogonal on a contour" Russian Math.Surveys , 21 : 2 (1966) pp. 35–83 Uspekhi Mat. Nauk , 21 : 2 (1966) pp. 41–88
[8] P.K. Suetin, "Polynomials orthogonal over a region and Bieberbach polynomials" Proc. Steklov Inst. Math. , 100 (1974) Trudy Mat. Inst. Steklov. , 100 (1971)

Comments

See also the state-of-the-art paper [a2] (on the theory) and [a1] (on digital signal processing applications).

References

[a1] Ph. Delsarte, Y. Genin, "On the role of orthogonal polynomials on the unit circle in digital signal processing applications" P. Nevai (ed.) , Orthogonal polynomials: theory and practice , Kluwer (1990) pp. 115–133
[a2] E.B. Saff, "Orthogonal polynomials from a complex perspective" P. Nevai (ed.) , Orthogonal polynomials: theory and practice , Kluwer (1990) pp. 363–393
How to Cite This Entry:
Orthogonal polynomials on a complex domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_polynomials_on_a_complex_domain&oldid=48077
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article