Nevanlinna-Pick interpolation by rational functions with a single pole inside the unit disk

We devise an efficient algorithm that, given points z1, . . . , zk in the open unit disk D and a set of complex numbers {fi,0, fi,1, . . . , fi,ni−1} assigned to each zi , produces a rational function f with a single (multiple) pole in D, such that f is bounded on the unit circle by a predetermined positive number, and its Taylor expansion at zi has fi,0, fi,1, . . . , fi,ni−1 as its first ni coefficients

Nevanlinna-Pick meromorphic interpolation: The degenerate case and minimal norm solutions

The Nevanlinna-Pick interpolation problem is studied in the class S(K) of meromorphic functions f with k poles inside the unit disk D and with parallel to f parallel to(L)infinity((T)) \u3c = 1. In the indeterminate case, the parametrization of all solutions is given in terms of a family of linear fractional transformations with disjoint ranges. A necessary and sufficient condition for the problem being determinate is given in terms of the Pick matrix of the problem. The result is then applied to obtain necessary and sufficient conditions for the existence of a meromorphic function with a given pole multiplicity which satisfies Nevanlinna-Pick interpolation conditions and has the minimal possible LOO-norm on the unit circle T. (C) 2008 Elsevier Inc. All rights reserved

The Transformation of Issai Schur and Related Topics in an Indefinite Setting

We review our recent work on the Schur transformation for scalar generalized Schur and Nevanlinna functions. The Schur transformation is defined for these classes of functions in several situations, and it is used to solve corresponding basic interpolation problems and problems of factorization of rational J-unitary matrix functions into elementary factors. A key role is played by the theory of reproducing kernel Pontryagin spaces and linear relations in these spaces

Non-Euclidian Metrics and the Robust Stabilization of Systems with Parameter Uncertainty

Abstract-This paper considers, from a complex function theoretic point of view, certain kinds of robust synthesis problems. In particular, we use a certain kind of metric on the disk (the "hyperbolic" metric) which allows us to reduce the problem of robust stabilization of systems with many types of real and complex parameter variations to an easily solvable problem in nowEuclidian geometry. It is shown that several apparently different problems can be treated in a unified general framework. A new result on the gain margin problem for multivariable plants is also given. Finally, we apply our methods to systems with real zero or pole variations. f? and D are well known to be. conformally equivalent

Training Neural Networks Using Reproducing Kernel Space Interpolation and Model Reduction

We introduce and study the theory of training neural networks using interpolation techniques from reproducing kernel Hilbert space theory. We generalize the method to Krein spaces, and show that widely-used neural network architectures are subsets of reproducing kernel Krein spaces (RKKS). We study the concept of "associated Hilbert spaces" of RKKS and develop techniques to improve upon the expressivity of various activation functions. Next, using concepts from the theory of functions of several complex variables, we prove a computationally applicable, multidimensional generalization of the celebrated Adamjan- Arov-Krein (AAK) theorem. The theorem yields a novel class of neural networks, called Prolongation Neural Networks (PNN). We demonstrate that, by applying the multidimensional AAK theorem to gain a PNN, one can gain performance superior to both our interpolatory methods and current state-of-the-art methods in noisy environments. We provide useful illustrations of our methods in practice

Operator monotone functions and L\"owner functions of several variables

We prove generalizations of L\"owner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone