Szegő polynomials

Szegő polynomials
Name	Szegő polynomials
Discipline	Mathematics
Field	Complex analysis; Orthogonal polynomials; Spectral theory
Introduced	1915
Introduced by	Gábor Szegő

Szegő polynomials are sequences of polynomials orthogonal on the unit circle with respect to a positive Borel measure; they play a central role in complex analysis, spectral theory and approximation theory. Constructed from a nontrivial probability measure on the unit circle, these polynomials furnish canonical orthonormal bases and connect to operator models, moment problems and scattering theory. Their structure is governed by recurrence relations, parameter sequences and deep asymptotic results with applications across mathematical physics and numerical analysis.

Definition and Basic Properties

Let μ be a nontrivial probability measure supported on the unit circle in the complex plane. The orthonormal Szegő polynomials are the sequence {φ_n(z)} with degree n satisfying ∫_{|z|=1} φ_n(z) \overline{φ_m(z)} dμ(z) = δ_{mn}. Existence and uniqueness derive from Gram–Schmidt applied to {1, z, z^2, ...} and relate to classical moment problems studied by Gábor Szegő, Szemerédi, Nikolai Bogolyubov, Andrei Kolmogorov, Émile Picard. Basic properties include monic normalization, conjugation relations with reversed polynomials, and zero distribution inside the unit disk connected to results by Marcel Riesz, Frigyes Riesz, John von Neumann, Bernhard Riemann, Hermann Weyl.

Orthogonality and Szegő Recurrence

Orthogonality leads to a three-term structure encoded in the Szegő recurrence, a first-order matrix recurrence connecting φ_{n+1}, φ_n and their reverse polynomials. The recurrence can be cast in terms of unitary transfer matrices related to models of Paul Dirac, Werner Heisenberg, Erwin Schrödinger, and appears in spectral decompositions studied by Marshall H. Stone and John von Neumann. The recurrence underpins numerical algorithms due to Richard Brent, Alan Turing, James H. Wilkinson, and connects with continued fraction expansions associated with Carl Friedrich Gauss, Adrien-Marie Legendre, Leonhard Euler.

Verblunsky Coefficients and Parametrization

Szegő polynomials are parametrized by the sequence of Verblunsky coefficients (also called reflection or Schur parameters), a bijection between nontrivial measures on the unit circle and sequences in the open unit disk. This parametrization was developed via work related to Simon and earlier insights by Otto Toeplitz, Isaac Schur, Thomas Stieltjes, and George Pólya. The Verblunsky map is central in inverse spectral problems studied by Mark Kac, Boris Levitan, Mikhail Gromov, and explicit factorization results relate to operator models by Bálint Szőkefalvi-Nagy, C. R. Putnam, N. I. Akhiezer.

Asymptotics and Szegő Theorems

Strong Szegő theorems describe asymptotics of Toeplitz determinants and polynomial norms in terms of logarithmic integrals of weight functions, with roots in work by Gábor Szegő, Harold Widom, Félix Berezin, Ludwig Faddeev. Extensions include results by Barry Simon linking to spectral theory of unitary operators, and connections to large n asymptotics in random matrix theory explored by Tracy Widom, Craig Tracy, Harold Widom, Percy Deift, Alexander Its. Szegő-type limit theorems interface with potential theory advances by Oded Schramm, John Garnett, Walter Rudin and with orthogonal polynomial asymptotics studied by Stieltjes, Hermite, Szegő's school.

Applications and Connections

Szegő polynomials appear in prediction theory for stationary stochastic processes pioneered by Norbert Wiener, Norbert Wiener's circle of influence, and in time-series analysis linked to methods by George Udny Yule, Maurice Kendall, Andrey Kolmogorov. They undergird filter design in engineering influenced by Claude Shannon, Harry Nyquist, Nyquist-type sampling theorems, and play roles in integrable systems related to work by Mikhail Zakharov, Peter Lax, Vladimir Zakharov, Lax pair formalisms. Connections extend to random matrix ensembles investigated by Eugene Wigner, Freeman Dyson, Mehta and to signal processing techniques used by Alan Oppenheim and Ronald Bracewell.

Examples and Explicit Families

Classical explicit families arise when the measure has particular weights: for the Lebesgue measure one obtains simple normalized monomials; for weights given by rational functions one obtains Geronimus and Bernstein–Szegő families tied to results by Ivan Niven, Nikolai Geronimus, S. Bernstein, G. Szegő. Measures supported on arcs lead to orthogonal systems connected to Chebyshev polynomials studied by Pafnuty Chebyshev and to Jacobi polynomials developed by Carl Gustav Jacobi. Explicit parameter choices produce paraorthogonal polynomials related to scattering matrices in works by L. D. Faddeev and M. G. Krein.

Category:Orthogonal polynomials