N-Dimensional Phase Approximation in the L_\infty -Norm

Abstract

The phase functions of N-dimensional (N-D) digital all-pass filters are investigated to approximate a prescribed phase response in a frequency region. The set of phase functions of the all-pass filters have common properties with some nonlinear approximating functions. This similarity answers the question of characterization of minimal approximation in the set of phase functions. The optimal approximation is characterized by known theorems of Tschebycheff Approximation Theory. Among the main tools of the theory, the Global and Local Kolmogoroff Criteria, are shown to give necessary and sufficient conditions for best approximations in the phase functions of N-D all-pass filters. Moreover, this best approximation in the phase functions is shown to be a global minimum. The approximation on discrete point sets ( H- sets) in a compact multidimensional domain is studied. Optimal N-dimensional approximation is not unique, an inherent property of functions of several variables.