Some factorizations of matrix functions in several variables. (English) Zbl 0784.15006

This paper seems to fire (almost) the parting shots in a struggle which, after venerable beginnings 250-90 years ago, was reviewed, on hand of an application to signal processing, a dozen of years ago by F. Neuman [Czech. Math. J. 32(107), 582-588 (1982; Zbl 0517.15012)] in that it offers necessary and sufficient conditions for nonsingular-matrix-valued functions on a cartesian product of arbitrary sets to be factorizable in the sense \[ H(x_ 1,x_ 2,\ldots,x_ m)=F_ 1(x_ 1)F_ 2(x_ 2)\ldots F_ m(x_ m). \] The bracketed “almost” above refers to the problem posed by the author, where \(x_ j\) on the right-hand side is replaced by a known function \(f_ j\) of \(x_ 1,x_ 2,\ldots,x_ m\) \((j=1,2,\ldots,m)\). He has a partial result for that case too and an example. Counter-examples disprove some tempting conjectures.


15A23 Factorization of matrices
15A54 Matrices over function rings in one or more variables
26B40 Representation and superposition of functions


Zbl 0517.15012
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