View source for Orthogonal polynomials on a complex domain

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.==
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:

<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>

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.

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>

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

<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>

Under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070350/o07035012.png" />, 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>

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" />.

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

<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>

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>

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.

==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

<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>

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" />.

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,

<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 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>

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>

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.

==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

<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>

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

<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>

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.

====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>



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.B. Saff,   "Orthogonal polynomials from a complex perspective"  P. Nevai (ed.) , ''Orthogonal polynomials: theory and practice'' , Kluwer  (1990)  pp. 363–393</TD></TR></table>