N-Dimensional Phase Approximation in the L∞-Norm

Thephase functions of N-dimensional (N-D) digital all-pass filtersare investigated to approximate a prescribed phase response ina frequency region. The set of phase functions of the all-passfilters have common properties with some nonlinear approximatingfunctions. This similarity answers the question of characterizationof minimal approximation in the set of phase functions. The optimalapproximation is characterized by known theorems of TschebycheffApproximation Theory. Among the main tools of the theory, theGlobal and Local Kolmogoroff Criteria, are shown to give necessaryand sufficient conditions for best approximations in the phasefunctions of N-D all-pass filters. Moreover, this best approximationin the phase functions is shown to be a global minimum. The approximationon discrete point sets (H-sets) in a compact multidimensionaldomain is studied. Optimal N-dimensional approximation is notunique, an inherent property of functions of several variables.