Rational Zernike functions

Abstract. Using the relation between the Poincaré and the Cayley–Klein models of hyperbolic geometry, and the congruence transformations on these models (closely related to Blaschke functions), we present a new hyperbolic implementation of Nelder–Mead simplex method, and construct novel complete orthonormal systems on the disk based on Zernike and Blaschke functions.

Key words and phrases. Rational Zernike functions, Nelder–Mead algorithm, Blaschke group, hyperbolic geometry.