Generalized interpolation in H-infinity with a complexity constraint

Researchers: Anders Lindquist in cooperation with C. I. Byrnes (Washington University, St Louis), T. T. Georgiou (University of Minnesota) and A. Megretski (MIT).

Sponsors: The Swedish Research Council (VR)  and the Göran Gustafsson Foundation .

In a seminal paper, Sarason generalized some classical interpolation problems for 23#23 functions on the unit disc to problems concerning lifting onto 24#24 of an operator 25#25 that is defined on 26#26 (27#27 is an inner function) and commutes with the (compressed) shift 28#28. In particular, he showed that interpolants (i.e., 29#29 such that 30#30) having norm equal to 31#31 exist, and that in certain cases such an 32#32 is unique and can be expressed as a fraction 33#33 with 34#34. In [A2], we study interpolants that are such fractions of 35#35 functions and are bounded in norm by 36#36 (assuming that 37#37, in which case they always exist). We parameterize the collection of all such pairs 38#38 and show that each interpolant of this type can be determined as the unique minimum of a convex functional.

Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory and signal processing, where 27#27 is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint. These are the problems we study in the project Rational Nevanlinna-Pick interpolation with degree constraints. We generalize this to the case of arbitrary inner functions by first constructing on a certain set a differential form which is exact (in an appropriate sence) and which gives rise intrincically to a convex optimization problem. Indeed, or method of proof reposes on a rigorous treatment of nonlinear optimization on certain (nonreflexive) Banach spaces.