Two observations regarding first-quadrant causal BIBO-stable digital filters

An n-dimensional digital filter is said to be normal if it is linear shift-invariant, first-quadrant causal, and bounded-input-bounded output (BIBO) stable. The transfer function of such a filter is holomorphic in the open polydisc Uⁿ and continuous in its closure, making it a member of W. Rudin´s polydisc algebra A(Uⁿ) (see W. Rudin) (Function Theory in Polydiscs, New York: W.A. Benjamin, 1969). It has been shown by Rudin that, nevertheless, every all-pass function in A(U ⁿ) is rational, free of nonessential singularities of the second kind, and of finite norm. This result appears to have been completely overlooked in the engineering literature. The material covered here is mainly tutorial, comprising a leisurely expanded account of the key ideas behind the Fejer and Rudin constructions