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).

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