*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.