Topics in Interpolation

College Of William And Mary, Williamsburg VA

Investigators

Abstract

Abstract Bolotnikov This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The proposal is primarily concerned with boundary interpolation problems for Schur-class functions (analytic self-maps of the unit disk) and their various generalizations. The PI will establish necessary and sufficient conditions for the existence of a Schur-class function with a prescribed jet (finite or infinite) of values for boundary derivatives at a given point on the unit circle. The main difficulty here is that in a generic case, the data set does not satisfy positivity conditions which are needed to apply the existing function- and operator-theoretic interpolation methods to full extent. These methods will be supplemented and combined with explicit constructions from geometric function theory. The project will also include implementable algorithms for constructing rational Schur-class functions of the minimally possible MacMillan degree with finitely many prescribed Taylor coefficients at a given boundary point. The construction will be used to study spectral theory of Fredholm operators in algebras generated by the shift operator and the composition operator with a Schur-class symbol. Another application will be found in complex dynamics: a similar construction carried out for positive real functions will eventually produce a one-parametric semigroup of composition operators with prescribed boundary behavior near the boundary Denjoy-Wolff point. The scalar-valued single-variable results will be extended in the following four directions: (1) to the multi-point case where the jets of boundary derivatives for the unknown interpolant are prescribed at finitely or countably many points on the unit circle; (2) to operator-valued Schur-class functions, whose values are contractive operators mapping one Hilbert space into another; (3) to meromorphic functions with the boundary values essentially bounded by one on the unit circle; (4) to various multivariable extensions of the Schur class including in particular, Schur-Agler functions of the polydisk and contractive multipliers of the Drury-Arveson space. The proposed research addresses the problems in the norm-constrained interpolation theory which has traditional connections with control theory, system theory, and signal and image processing. Besides obtaining concrete results, the objective of this research is to advance the understanding of the interaction of operator theory and complex function theory in the present boundary framework. The multivariable case is still much less studied and therefore, is of particular interest. To study this case, the functional-model canonical realizations for Schur-Agler functions will be constructed, which in turn will establish the bridge to multivariable system theory as well as to model theory for various tuples of Hilbert space operators. The results obtained in this project will be published in scientific journals and reported at research conferences. The project contains several problems reserved for undergraduate students who will be engaged in intensive research by the PI.