Belgisch wiskundig genootschap / Société mathématique de Belgique
Tijdschrift van het Belgisch Wiskundig Genootschap Reeks B / Bulletin de la Société Mathématique de Belgique Serie B
The relation between the Wiener-Masani theory of multivariable prediction and the multivariable Szegö theory of orthogonal polynomials recently studied by Delsarte et al. and Youla et al. is well understood. The classical Levinson and Schur algorithms are based on this theory and produce autoregressive filters by ‘incoming’ and ‘outgoing’ recursions, respectively. The Nevanlinna-Pick algorithm generalizes the Schur algorithm to produce ARMA filters. In this note, the corresponding incoming algorithm is studied which gives recursively reproducing kernels and orthogonal functions in a Hilbert module, thus building a resolution of submodules, constructed on the transmission zeros of the filter. A survey of results that were obtained in the author’s doctoral thesis is given, without going into the details of the proofs. This is the multivariable extension of a similar note [A. Bultheel and P. Dewilde, International Symposium on the Mathematical Theory of Networks and Systems, Vol. 3 (Delft, 1979), 207-212, Western Periodicals, North Hollywood, Calif., 1979] which treats the scalar case.