Alexander Kheifets

The Higher Order Carathéodory—Julia Theorem and Related Boundary Interpolation Problems

The higher order analogue of the classical Caratheodory-Julia theorem on boundary angular derivatives has been obtained in [7]. Here we study boundary interpolation problems for Schur class functions (analytic and bounded by one in the open unit disk) motivated by that result.

Boundary Nevanlinna—Pick Interpolation Problems for Generalized Schur Functions

Three boundary multipoint Nevanlinna-Pick interpolation problems are formulated for generalized Schur functions. For each problem, the set of all solutions is parametrized in terms of a linear fractional transformation with a Schur class parameter.

Nevanlinna-Pick interpolation: Pick matrices have bounded number of negative eigenvalues

The Nevanlinna-Pick interpolation problem is studied in the class of functions defined on the unit disk without a discrete set, with the property that all their Pick matrices have not more than a prescribed number of negative eigenvalues. It is shown, in particular, that the degenerate problem always has a unique solution, not necessarily meromorphic. A related extension problem to a maximal function in the class is also studied.

On a necessary but not sufficient condition for a γ-generating pair to be a nehari pair

This paper gives a negative answer to a question due to V.M. Adamjan, D.Z. Arov and M.G. Krein, and (what is the same) gives a counterexample to D.Sarasons conjecture* concerning exposed points inH1.

Confident Predictability: Identifying reliable gene expression patterns for individualized tumor classification using a local minimax kernel algorithm

BackgroundMolecular classification of tumors can be achieved by global gene expression profiling. Most machine learning classification algorithms furnish global error rates for the entire population. A few algorithms provide an estimate of probability of malignancy for each queried patient but the degree of accuracy of these estimates is unknown. On the other hand local minimax learning provides such probability estimates with best finite sample bounds on expected mean squared error on an individual basis for each queried patient. This allows a significant percentage of the patients to be identified as confidently predictable, a condition that ensures that the machine learning algorithm possesses an error rate below the tolerable level when applied to the confidently predictable patients.ResultsWe devise a new learning method that implements: (i) feature selection using the k-TSP algorithm and (ii) classifier construction by local minimax kernel learning. We test our method on three publicly available gene expression datasets and achieve significantly lower error rate for a substantial identifiable subset of patients. Our final classifiers are simple to interpret and they can make prediction on an individual basis with an individualized confidence level.ConclusionsPatients that were predicted confidently by the classifiers as cancer can receive immediate and appropriate treatment whilst patients that were predicted confidently as healthy will be spared from unnecessary treatment. We believe that our method can be a useful tool to translate the gene expression signatures into clinical practice for personalized medicine.

Pairs of functions with indefinite Pick matrices

Results of factorization type are proved that characterize pairs of functions whose Pick matrices have not more than a prescribed number of negative eigenvalues. These results are in turn used to describe functions having Caratheodory matrices with bounded number of negative eigenvalues.

On Some Special Cases of the Radon–Nikodym Theorem for Vector- and Operator-valued Measures

This paper presents a proof of the Radon–Nikodym theorem for vector measures with values in a Hilbert space or in the space of bounded linear operators acting from a Hilbert space to a Hilbert space. Assertions for these cases are known ([13], [14], [15]), however they contain some mistakes and inaccuraces (see Concluding Remarks at the end of this paper). Considering operator-valued measures, we emphasize distinctions between the uniform and strong topologies (see Remark 2.3 to Lemma 2.2, Theorem 2.5 and Corollary 2.6). There exist more general versions of the Radon–Nikodym theorem: for measures with values in Banach spaces with boundedly complete Schauder basis or for separable dual Banach spaces (detailed exposition and history can be found, e.g., in [10]). However, we think that a direct and simple proof for the Hilbert space case is of independent interest.

On the scattering problem in Ryckman's class of Jacobi matrices

We give a complete solution of the scattering problem for Jacobi matrices from a class which was recently introduced by E. Ryckman. We characterize the scattering data for this class and illustrate the inverse scattering on some simple examples.