Szegö polynomial

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

where is a positive measure on (cf. also Orthogonal polynomials on a complex domain). The monic orthogonal Szegö polynomials satisfy a recurrence relation of the form

for , with initial conditions and . Here, if . The parameter is called a reflection coefficient or Schur or Szegö parameter.

Szegö's extremum problem is to find , with the -norm and where the minimum is taken over all ( being the open unit disc) satisfying . If is restricted to be a polynomial of degree at most , then a solution is given by .

Szegö's theory involves the solution of this extremum problem and related questions such as the asymptotics of as . The essential result is that equals the geometric mean of , i.e., with . Szegö's condition is that , and it is equivalent with and with the fact that the system is not complete in (cf. also Complete system).

Defining the orthonormal Szegö polynomials

then if Szegö's condition holds one has

where the Szegö function is defined as

with the Riesz–Herglotz kernel (cf. also Carathéodory class). The convergence holds uniformly on compact subsets . The function is an outer function (cf. Hardy classes) in with radial limit to the boundary, and a.e. . Therefore it is also called a spectral factor of the weight function . Other asymptotic formulas were obtained under much weaker conditions, such as a.e. or the Carleman conditions for the moments of .

Szegö polynomials of the second kind are defined inductively as and, for ,

The rational functions interpolate the Riesz–Herglotz transform

at zero and infinity. 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 and a Schur function (cf. also Schur functions in complex function theory), namely

A Schur function is analytic and its modulus is bounded by in . I. Schur developed a continued-fraction-like algorithm to extract the reflection coefficients from . It is based on the recursive application of the lemma saying that is a Schur function if and only if and

is a Schur function. The correspond to reflection coefficients associated with if and the successive approximants that are computed for are related to the Cayley transforms of the interpolants given above. It also follows that there is an infinite sequence of reflection coefficients in , unless is a rational function, i.e. unless is a discrete measure. It also implies that, except for the case of a discrete measure, the Szegö polynomials have all their zeros in .

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., , using an appropriate transformation. Given the polynomials orthogonal for a weight function on an interval , then the orthogonal polynomials for a rational modification , where is a polynomial positive on , can be derived. Bernshtein–Szegö polynomials are orthogonal polynomials for rational modifications of one of the four classical Chebyshev weights on , i.e. for with .

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)