Section: New Results

Orthogonal rational functions and non-stationary stochastic processes

Participants : Laurent Baratchart, Stanislas Kupin [Univ. Bordeaux 1] .

The theory of orthogonal polynomials on the unit circle is a most classical piece of analysis which is still the object of intensive studies. The asymptotic behaviour of orthogonal polynomials is of special interest for many issues pertaining to approximation theory and to spectral theory of differential operators. Its connection with prediction theory of stationary stochastic processes has long been known [64] . Namely, the $n$-th orthonormal polynomial with respect to the spectral measure of the process yields the optimal regression coefficients of a linear one-step ahead predictor from the $n-1$-st last values, in the sense of minimum variance of the error. Likewise, the (inverse of) the dominant coefficient of the polynomial gives the prediction error. In particular, asymptotics for the dominant coefficient determine the asymptically optimal prediction error from the past as time goes large.

As compared to orthogonal polynomials, orthogonal rational functions have not been much considered up to now. They were apparently introduced by Dzrbasjan but the first systematic exposition seems to be the monograph by Bultheel et al. [57] where the emphasis is more on the algebraic side of the theory. In fact, the asymptotic analysis of orthogonal rational functions is still in its infancy.

We recently developed an analog of the Kolmogorov-Krein-Szegö theorem [18] for orthogonal rational functions which is first of its kind in that it allows for the poles of these functions to approach the unit circle, generalizing previously known results for compactly supported singular set. Dwelling on this asymptotic analysis of orthogonal rational functions, we developed a prediction theory for certain, possibly nonstationary stochastic processes that we call Blaschke varying processes. [44] . These are characterized by a spectral calculus where time shift corresponds to multiplication by an elementary Blaschke product (that may depend on the time instant considered). This class of processes contains the familiar Gaussian stationary processes, but it contains many more that exhibit a much more varied behaviour. For instance, the process may be asymptotically deterministic along certain subsequences and nondeterministic along others. The optimal predictor is constructed from the spectral measure via orthogonal rational functions, and its asymptotic behaviour is characterized by the above-mentioned generalization of the Kolmogorov-Krein-Szegö theorem. In the same vein, we also developed prediction theory for another class of nonstationary processes, the so-called Cauchy-processes, that may be characterized as stationary processes feedind in turn a sequence of varying filters of degree 1. Their covariance matrices can be charatcerized via Nevanlinna-Pick interpolation. The issue of characterizing covariance sequences of Blaschke processes is still open. Their identification raises the problem of constructing optimal Schur rational approximants to a given Schur function.