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Department of Mathematics
A short course on Nevanlinna-Pick interpolation theory
General information
A minicourse entitled A short course on Nevanlinna-Pick interpolation
theory is to be held at the Åbo Akademi University, Department of
Mathematics, in Åbo, Finland, November 26-29, 2001. Two international
experts Prof. Joe Ball (Virginia Tech.) and Prof. Anders Lindquist (KTH) will
give a minicourse suitable for graduate students with some basic knowlegde of
complex analysis and functional analysis. The course (16 h) can be taken for
credits. There will be no registration fee. A small number of rooms have been
reserved for the participants at Domus Aboensis, Biskopsgatan 10, 20500 Åbo.
The charge is 120-150 mk for one night. Those interested in participating in
the minicourse are asked to fill in and send us the registration form by
30.9.2001. For further information please contact the organizers Mikael
Lindström and Olof Staffans.
Abstract of lectures by Joe Ball
The lectures will trace the development of the theory of
Nevanlinna-Pick from the simplest classical formulation through more recent
bitangential matrical versions developed in the 1980s (by both mathematicians
and engineers) for applications to emerging H-infinity control theory.
Connections of the problem and its solution with system theory ideas will be
emphasized. We shall also
point out a recent unified abstract formulation of the problem closely
associated with the geometry of a Lax-Phillips scattering system. More recent
generalizations to time-varying system and various types of multidimensional
systems along with their applications to other types of robust control
problems will also be highlighted.
Abstract of lectures by Anders Lindquist
Many important problems in circuit theory, robust stabilization and control,
signal processing, speech synthesis, and stochastic systems theory can
be formulated as analytic interpolation problems. For these problems the
(real) interpolants are (i) positive real, i.e., analytic in the unit disc (or
its complement) and mapping it into the closed left-half plane, and (ii)
rational of a prescribed maximal degree. Removing the degree requirement (ii),
this is a classical problem going back to Schur, Caratheodory, Toeplitz,
Nevanlinna and Pick. However, the degree constraint is essential in
applications, and it
considerably alters the mathematical problem. In fact, it has been shown
only recently that, if there are $n+1$ interpolation points, there is a
complete parameterization of all interpolants of degree at most $n$ in terms of
spectral zeros, which hence become design parameters. Each such solution can be
determined by solving a convex optimization problem.
This series of lectures will cover the main aspects of the theory of
analytic interpolation with degree constraint, and the theory will be
illustrated by examples from robust control and high-resolution spectral
estimation. We will begin by considering a special case formulated by
Kalman, the rational covariance extension problem, in the context of speech
processing. Then, certain manifolds and submanifolds of positive real transfer
functions will be studied, and they will be motivated in terms of both analytic
interpolation and a fast algorithm for Kalman filtering, viewed as a
nonlinear dynamical system on the space of positive real transfer functions and
related to spectral factorization. In particular, a fundamental geometric
duality between filtering and analytic interpolation will be described.
This duality, while interesting in its own right, has several corollaries
which provide solutions
and insight into some very interesting and intensely researched
problems. In the context of signal processing, this leads to the global
analysis of linear systems, where one studies an entire class of systems or
models as a whole, and where one views measurements, such as covariance lags
and cepstral coefficients or Markov parameters, from data as functions on
the entire class. This allows us to give rigorous answers to many modeling
problems in an algorithm-independent fashion.
Finally, it will be shown that each interpolant can be obtained as the
solution to a problem of maximizing a generalized entropy functional.
These optimization problems have duals which are convex optimization problems
in a finite-dimensional space. This leads to efficient computational
procedures.