On reproducing kernel Riesz bases in model spaces

Reproducing kernel Riesz bases associated to real points in modelsubspaces $K_I^2= H^2ominus IH^2$ of the Hardy space $H^2$ havebeen described in terms of equality of spaces by A. Baranovcite{Ba2}. We now answer the question of whether the natural analogue of his result withthe real points being replaced by sequences of points located in thewhole complex plane holds. It turns out that the analogous conditions areindeed sufficient but not in general necessary. A consequence ofthis is that larger perturbations of such bases are admissible withsequences of associated points from the upper or lower half-planethan along the real line.