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[PAST EVENT] Mathematics Colloquium: Maxym Derevyagin (The University of Mississippi)
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- Open to the public
Speaker: Maxym Derevyagin (The University of Mississippi)
Title: On linear pencils of Jacobi matrices and Nevanlinna-Pick problems
Abstract: We will begin by discussing relations between a few classical objects such as orthogonal polynomials, Nevanlinna functions, Hamburger moment problems, Jacobi matrices, and Pad? approximation. The latter is a very efficient and yet mysterious tool in computational mathematics, which, for instance, is very effective in extracting information about singularities of meromorphic functions from their Taylor coefficients, aka moments. However, some mathematical models deal with probability measures that do not necessarily have finite moments of all orders and, therefore, the Pad? approximation scheme cannot directly be applied to the underlying analytic function. Nevertheless, there is a way to overcome this obstacle by dealing with an interpolation problem for the Cauchy transform of a measure rather than a moment problem. So, the goal of the second part of the talk is to present the theory of interpolating fractions from the spectral point of view. It?ll be shown that starting with the interpolation data one can get generalized eigenvalue problems that involve two Jacobi matrices. It turns out that the corresponding eigenvectors have orthogonal rational functions as entries. In other words, the underlying spectral object is a linear pencil of two Jacobi matrices and such pencils can be used to study Nevanlinna-Pick problems. In particular, the uniqueness criteria for solutions of Nevanlinna-Pick problems will be presented.
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Vladimir Bolotnikov