Definition
Suppose that is a probability measure on the unit circle in the complex plane, whose support is not finite. The orthogonal polynomials associated to are the polynomials with leading term that are orthogonal with respect to the measure .The Szegő recurrence
Szegő's recurrence states that : : where : is the polynomial with its coefficients reversed and complex conjugated, and where the Verblunsky coefficients are complex numbers with absolute values less than 1.Verblunsky's theorem
Verblunsky's theorem states that any sequence of complex numbers in the open unit disk is the sequence of Verblunsky coefficients for a unique probability measure on the unit circle with infinite support.Geronimus's theorem
Geronimus's theorem states that the Verblunsky coefficients of the measure μ are the Schur parameters of the function defined by the equations :Baxter's theorem
Baxter's theorem states that the Verblunsky coefficients form an absolutely convergent series if and only if the moments of form an absolutely convergent series and the weight function is strictly positive everywhere.Szegő's theorem
Szegő's theorem states that : where is the absolutely continuous part of the measure .Rakhmanov's theorem
Rakhmanov's theorem states that if the absolutely continuous part of the measure is positive almost everywhere then the Verblunsky coefficients tend to 0.Examples
The Rogers–Szegő polynomials are an example of orthogonal polynomials on the unit circle.References
* * * * * *{{Citation , last1=Szegő , first1=Gábor , title=Orthogonal Polynomials , url=https://books.google.com/books?id=3hcW8HBh7gsC , publisher= American Mathematical Society , series=Colloquium Publications , isbn=978-0-8218-1023-1 , mr=0372517 , year=1939 , volume=XXIII Orthogonal polynomials