Entropy and multivariable interpolation

Entropy and Multivariable Interpolation

We define a new notion of entropy for operators on Fock spaces and positive definite multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive definite multi-Toeplitz kernels on free semigroups (resp. multi-T...

Multivariable Interpolation Problems

Multivariable Operator-valued Nevanlinna-pick Interpolation: a Survey

The theory of Nevanlinna-Pick and Carathéodory-Fejér interpolation for matrixand operator-valued Schur class functions on the unit disk is now well established. Recent work has produced extensions of the theory to a variety of multivariable settings, including the ball and the polydisk (both commutative and noncommutative versions), as well as a time-varying analogue. Largely independent of thi...

Improved interpolation inequalities, relative entropy and fast diffusion equations

We consider a family of Gagliardo-Nirenberg-Sobolev interpolation inequalities which interpolate between Sobolev’s inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder t...

Sensitivity Shaping under Degree Constraint: Nevanlinna-Pick Interpolation for Multivariable and Time-Delay Systems

It is well known that analytic interpolation theory has found various applications in systems and control, e.g., H∞ control, covariance extension problem for spectral estimation, filtering and gain equalization. For the engineering point of views, it is desirable to synthesize interpolants with bounded degree, where the interpolants correspond to the transfer functions, representing devices of ...