Szegö polynomial

The Szegö polynomials form an orthogonal polynomial sequence with respect to the positive definite Hermitian inner product

\begin{equation*} \langle f , g \rangle = \int _ { - \pi } ^ { \pi } f ( e ^ { i \theta }) \overline { g ( e ^ { i \theta } ) } d \mu ( \theta ), \end{equation*}

where $\mu$ is a positive measure on $[ - \pi , \pi )$ (cf. also Orthogonal polynomials on a complex domain). The monic orthogonal Szegö polynomials satisfy a recurrence relation of the form

\begin{equation*} \Phi _ { n + 1 } ( z ) = z \Phi _ { n } ( z ) + \rho _ { n + 1 } \Phi _ { n } ^ { * } ( z ), \end{equation*}

for $n \geq 0$, with initial conditions $\Phi _ { 0 } = 1$ and $\Phi _ { - 1 } ( z ) = 0$. Here, $\Phi _ { n } ^ { * } ( z ) = \sum _ { k = 0 } ^ { n } \overline { b } _ { n k } z ^ { n - k }$ if $\Phi _ { n } ( z ) = \sum _ { k = 0 } ^ { n } b _ { n k } z ^ { k }$. The parameter $\rho _ { n + 1} = \Phi _ { n + 1 } ( 0 )$ is called a reflection coefficient or Schur or Szegö parameter.

Szegö's extremum problem is to find $\delta _ { \mu } = \operatorname { min } _ { H } \| H \| _ { \mu }$, with $\| H \| _ { \mu }$ the $L ^ { 2 } ( \mu )$-norm and where the minimum is taken over all $H \in H ^ { 2 } ( \mu , {\bf D} )$ ($\mathbf D$ being the open unit disc) satisfying $H ( 0 ) = 1$. If $H$ is restricted to be a polynomial of degree at most $n$, then a solution is given by $H = \Phi _ { n } ^ { * }$.

Szegö's theory involves the solution of this extremum problem and related questions such as the asymptotics of $\Phi _ { n } ^ { * }$ as $n \rightarrow \infty$. The essential result is that $\delta _ { \mu }$ equals the geometric mean of $\mu ^ { \prime }$, i.e., $\delta _ { \mu } = \operatorname { exp } \{ c _ { \mu } / ( 4 \pi ) \}$ with $c _ { \mu } = \int _ { - \pi } ^ { \pi } \operatorname { log } \mu ^ { \prime } ( \theta ) d \theta$. Szegö's condition is that $ { c } _ { \mu } > - \infty$, and it is equivalent with $\delta _ { \mu } > 0$ and with the fact that the system $\{ \Phi _ { k } \} _ { k = 0 } ^ { \infty }$ is not complete in $H ^ { 2 } ( \mu )$ (cf. also Complete system).

Defining the orthonormal Szegö polynomials

\begin{equation*} \phi _ { n } ( z ) = \frac { \Phi _ { n } ( z ) } { \| \Phi _ { n } \| _ { \mu } }, \end{equation*}

then if Szegö's condition holds one has

\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \phi _ { n } ^ { * } ( z ) = D _ { \mu } ( z ) ^ { - 1 }, \end{equation*}

where the Szegö function is defined as

\begin{equation*} D _ { \mu } ( z ) = \operatorname { exp } \left\{ \frac { 1 } { 4 \pi } \int _ { - \pi } ^ { \pi } \operatorname { log } \mu ^ { \prime } ( \theta ) R ( e ^ { i \theta } , z ) d \theta \right\}, \end{equation*}

with $R ( t , z ) = ( t + z ) / ( t - z )$ the Riesz–Herglotz kernel (cf. also Carathéodory class). The convergence holds uniformly on compact subsets $\mathbf D$. The function $D$ is an outer function (cf. Hardy classes) in $\mathbf D$ with radial limit to the boundary, and a.e. $| D _ { \mu } ( e ^ { i \theta } ) | ^ { 2 } = \mu ^ { \prime } ( \theta )$. Therefore it is also called a spectral factor of the weight function $\mu ^ { \prime }$. Other asymptotic formulas were obtained under much weaker conditions, such as $\mu ^ { \prime } > 0$ a.e. or the Carleman conditions for the moments of $\mu$.

Szegö polynomials of the second kind are defined inductively as $\psi _{0} = 1$ and, for $n \geq 1$,

\begin{equation*} \psi _ { n } ( z ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } R ( e ^ { i \theta } , z ) [ \phi _ { n } ( e ^ { i \theta } ) - \phi _ { n } ( z ) ] d \mu ( \theta ). \end{equation*}

The rational functions $F _ { n } = - \psi _ { n } / \phi _ { n }$ interpolate the Riesz–Herglotz transform

\begin{equation*} F _ { \mu } ( z ) = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } R ( e ^ { i \theta } , z ) d \mu ( \theta ) \end{equation*}

at zero and infinity. $F _ { \mu }$ is a Carathéodory or positive real function because it is analytic in the open unit disc and has positive real part there.

The Cayley transform gives a one-to-one correspondence between $F _ { \mu }$ and a Schur function (cf. also Schur functions in complex function theory), namely

\begin{equation*} S _ { \mu } ( z ) = \frac { F _ { \mu } ( z ) - F _ { \mu } ( 0 ) } { F _ { \mu } ( z ) + F _ { \mu } ( 0 ) }. \end{equation*}

A Schur function is analytic and its modulus is bounded by $1$ in $\mathbf D$. I. Schur developed a continued-fraction-like algorithm to extract the reflection coefficients from $S _ { \mu }$. It is based on the recursive application of the lemma saying that $S _ { k }$ is a Schur function if and only if $S _ { k } ( 0 ) \in \mathbf{D}$ and

\begin{equation*} S _ { k + 1 } ( z ) = z ^ { - 1 } \frac { S _ { k } ( z ) - S _ { k } ( 0 ) } { 1 - \overline { S _ { k } ( 0 ) }S _ { k } ( z ) } \end{equation*}

is a Schur function. The $S _ { k } ( 0 )$ correspond to reflection coefficients associated with $\mu$ if $S _ { 0 } = S _ { \mu }$ and the successive approximants that are computed for $S _ { \mu }$ are related to the Cayley transforms of the interpolants $F _ { n }$ given above. It also follows that there is an infinite sequence of reflection coefficients in $\mathbf D$, unless $S _ { \mu }$ is a rational function, i.e. unless $\mu$ is a discrete measure. It also implies that, except for the case of a discrete measure, the Szegö polynomials have all their zeros in $\mathbf D$.

All these properties have a physical interpretation and are important for the application of Szegö polynomials in linear prediction, modelling of stochastic processes, scattering and circuit theory, optimal control, etc.

The polynomials orthogonal on a circle are of course related to polynomials orthogonal on the real line or on an interval, e.g., $I = [ - 1,1 ]$, using an appropriate transformation. Given the polynomials orthogonal for a weight function $w$ on an interval $I$, then the orthogonal polynomials for a rational modification $w / p$, where $p$ is a polynomial positive on $I$, can be derived. Bernshtein–Szegö polynomials are orthogonal polynomials for rational modifications of one of the four classical Chebyshev weights on $I$, i.e. for $w ( x ) = ( 1 - x ) ^ { \alpha } ( 1 + x ) ^ { \beta }$ with $\alpha , \beta \in \{ - 1 / 2,1 / 2 \}$.

References

[a1]	G. Freud, "Orthogonal polynomials" , Pergamon (1971)
[a2]	Ya. Geronimus, "Orthogonal polynomials" , Consultants Bureau (1961) (In Russian)
[a3]	H. Stahl, V. Totik, "General orthogonal polynomials" , Encycl. Math. Appl. , Cambridge Univ. Press (1992)
[a4]	G. Szegö, "Orthogonal polynomials" , Colloq. Publ. , 33 , Amer. Math. Soc. (1967) (Edition: Third)