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

• Published in: Journal of computational and applied mathematics Vol. 179; no. 1; pp. 157 - 173
• Main Authors: Cruz-Barroso, Ruymán, González-Vera, Pablo
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Amsterdam Elsevier B.V 01.07.2005; Elsevier
• Subjects: Christoffel–Darboux formula; Exact sciences and technology; Favard's theorem; Fourier analysis; Mathematical analysis; Mathematics; Orthogonal Laurent polynomials on the unit circle; Sciences and techniques of general use; Special functions; Three-term recurrence relations; Three-term recurrence relations; Favard's theorem; Orthogonal Laurent polynomials on the unit circle; Christoffel–Darboux formula; Transcendental equation; Christoffel-Darboux formula; Recurrence relation; Unit circle; Laurent polynomial; Difference equation; Favard theorem; Orthogonal polynomial; Christoffel Darboux formula; Numerical analysis; Applied mathematics
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2004.09.039

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.