\(N\)-dimensional phase approximation in the \(L_\infty\)-norm (Q1590497)

A commonly used technique to approximate a prescribed phase in a frequency region consists of, according to a desired phase response, seeking the coefficients of an \(N\)-dimensional digital all-pass filter such that the maximum design error is minimized in the frequency region. Extending a work of the first author concerning the one-dimensional case [\textit{W. S. Kafri}, Signal Process. 57, No. 2, 163-175 (1997)] and making use of the theory of nonlinear Chebyshev approximation, multidimensional phase function properties are investigated. The characterization of the best approximation as a global minimum is emphasized. The approximation on discrete point sets in a compact multidimensional domain is also considered and several illustrative examples included.