# 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

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