A Christoffel–Darboux formula and a Favard's theorem for orthogonal Laurent polynomials on the unit circle

Abstract

Let { ϕ k ( z ) } k = 0 ∞ be the family of orthonormal Laurent polynomials on the unit circle which spans Δ in the “ordering” induced by p ( n ) = E [ ( n + 1 ) / 2 ] . From the three-term recurrence relation satisfied by { ϕ k ( z ) } k = 0 ∞ we deduce a Christoffel–Darboux formula. Particular examples are considered and a Favard-type theorem is proved. A connection with the ordering induced by p ( n ) = E [ n / 2 ] is also established.