Functional models and asymptotically orthonormal sequences
by
Emmanuel Fricain
Institut G. Desargues, Université Lyon I (FRANCE)
Coauthors: Isabelle Chalendar (IGD, Université Lyon I), Dan Timotin (Institute of Mathematics of the Romanian Academy, Bucharest)

A sequence (x_n)_{n >= 1} is an asymptotically orthonormal sequence in an Hilbert space H if there exists N₀ in N, such that for all N >= N₀, there are constants c_N, C_N > 0 verifying

cN å n >= N  |an|2 <= || å n >= N  anxn ||2 <= CN å n >= N  |an|2,

and lim_{N --> \infty} c_N = 1 = lim_{N --> \infty} C_N.

In 1982, Volberg characterized sequences of reproducing kernels of H² (the Hardy space of the unit disk) which form an asymptotically orthonormal sequence in terms of a strengthened Carleson condition. We study the case of reproducing kernels in the model space K_\Theta=H²\ominus \ThetaH², where \Theta is an inner function. In particular, we prove some stability results and discuss the important case of exponentials.

Date received: June 11, 2002