On the well-posedness of the rational covariance extension problem

Abstract. In this paper, we give a new proof of the solution of the rational covariance extension problem, an interpolation problem with historical roots in potential theory, and with recent application in speech synthesis, spectral estimation, stochastic systems theory, and systems identification. The heart of this problem is to parameterize, in useful systems theoretical terms, all rational, (strictly) positive real functions havinga specified window of Laurent coefficients and a bounded degree. In the early 1980’s, Georgiou used degree theory to show, for any fixed “Laurent window”, that to each Schur polynomial there exists, in an intuitive systems-theoretic manner, a solution of the rational covariance extension problem. He also conjectured that this solution would be unique, so that the space of Schur polynomials would parameterize the solution set in a very useful form. In a recent paper, this problem was solved as a corollary to a theorem concerning the global geometry of rational, positive real functions. This corollary also asserts that the solutions are analytic functions of the Schur polynomials. After giving an historical motivation and a survey of the rational covariance extension problem, we give a proof that the rational covariance extension problem is well-posed in the sense of Hadamard, i.e a proof of existence, uniqueness and continuity of solutions with respect to the problem data. While analytic dependence on the problem data is stronger than continuity, this proof is much more streamlined and also applies to a broader class of nonlinear problems. The paper concludes with a discussion of open problems. 1

Consilience for Universal Design : The Emergence of a Third Culture

Consilience offers a powerful mechanism for borrowing from other disciplines, thereby extending the scope of what can be known. This paper looks at the foundations of the activity of design as a means of providing IT systems that cater for diverse needs. Developing systems that are expected to satisfy needs continuously (i.e., systems that are expected to evolve) calls for a dynamic activity of design that is responsive to changes in the environment. The contrast with the scientific assumption of ordered development invokes a new classification, supported by insights from other disciplines that place the range of approaches in context. Design is neither orderly nor linear; it implies a continuous and active search to resolve trade-offs and satisfy changing constraints. The paper concludes by making a case for design as an alternative culture that borrows from, and supports, both the scientific and the literary cultures. Acceptance of the role of practice as an interface alongside the more traditional cultures enables researchers and practitioners to access and adopt a larger variety of methods and general approaches underpinning an even larger corpus of insights. Consilience can occur at different levels and offer a variety of benefits. Balancing scientific enquiry with artistic flair and creativity through careful sensemaking that supports sharing across cultures provides the greatest benefit from learning to look across rather than blindly focusing inwPeer reviewe