## 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.