The Commutant Lifting Approach to Interpolation Problems

ISBN-13: 9783034877145 / Angielski / Miękka / 2013 / 632 str.

Foias;Frazho

Opis

Classical H interpolation theory was conceived at the beginning of the century by C. Caratheodory, L. Fejer and I. Schur. The basic method, due to Schur, in solving these problems consists in applying the Mobius transform to peel off the data. In 1967, D. Sarason encompassed these classical interpolation problems in a representation theorem of operators commuting with special contractions. Shortly after that, in 1968, B. Sz.- Nagy and C. Foias obtained a purely geometrical extension of Sarason's results. Actually, their result states that operators intertwining restrictions of co-isometries can be extended, by preserving their norm, to operators intertwining these co-isometries; starring with R. G. Douglas, P. S. Muhly and C. Pearcy, this is referred to as the commutant lifting theorem. In 1957, Z. Nehari considered an L interpolation problern which in turn encompassed the same classical interpolation problems, as well as the computation of the distance of a function f in L to H . At about the sametime as Sarason's work, V. M.

Spis treści

I. Analysis of the Caratheodory Interpolation Problem.- II. Analysis of the Caratheodory Interpolation Problem for Positive-Real Functions.- III. Schur Numbers, Geophysics and Inverse Scattering Problems.- IV. Contractive Expansions on Euclidian and Hilbert Space.- V. Contractive One Step Intertwining Liftings.- VI. Isometric and Unitary Dilations.- VII. The Commutant Lifting Theorem.- VIII. Geometric Applications of the Commutant lifting Theorem.- IX. H? Optimization and Functional Models.- X. Some Classical Interpolation Problems.- XI. Interpolation as a Momentum Problem.- XII. Numerical Algorithms for H? Optimization in Control Theory.- XIII. Inverse Scattering Algorithms for the Commutant Lifting Theorem.- XIV. The Schur Representation.- XV. A Geometric Approach to Positive Definite Sequences.- XVI. Positive Definite Block Matrices.- XVII. A Physical Basis for the Layered Medium Model.- References.- Notation.