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 having positive leading coefficient and satisfying the orthogonality (usually orthonormality) condition:

where is a bounded non-decreasing function on the interval with an infinite number of points of growth, called a distribution function, while is the Kronecker symbol. A recurrence relation and the analogue of the Christoffel–Darboux formula holds for the polynomials , in the same way as in the case of orthogonality on an interval.

Asymptotic properties are examined under the condition

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 be orthonormal on the segment with differential weight function , and let the weight function on the circle have the form

Under the condition , the Szegö formula

holds, where is the leading coefficient of the polynomial .

If an analytic function in the disc has non-tangential boundary values on the circle , then under certain supplementary conditions the expansion

(1)

holds; its coefficients are defined by the formula

Series of the form (1) are direct generalizations of Taylor series: if , . Given certain conditions on the distribution function , the series (1) converges or diverges simultaneously with the Taylor series of the same function at the points of the circle , i.e. the theorem on equiconvergence of these two series holds.

Orthogonal polynomials over a contour.

A system of polynomials having positive leading coefficient and satisfying the condition

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

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

a Fourier series in the orthogonal polynomials,

(2)

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

is fulfilled, where the infimum is taken over the set of all polynomials , then the series (2) converges in the mean to the function along the contour with weight and, under certain supplementary conditions, inside the domain as well.

Orthogonal polynomials over a domain.

A system of polynomials having positive leading coefficient and satisfying the condition

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

is fulfilled, where the infimum is taken over the set of all polynomials , then the Fourier series in the polynomials of an analytic function in a simply-connected domain converges in the mean (with respect to the area of the domain ) with weight to this function and, under certain supplementary conditions, inside the domain 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