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Hyponormal quantization of planar domains

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CATW01 - The complex analysis toolbox: new techniques and perspectives

By replacing the identity operator in Heisenberg commutation relation
[T*,T]=I by a rank-one projection one unveils an accessible spectral analysis classification
with singular integrals of Cauchy type as generic examples. An inverse spectral problem for this class
of (hyponormal) operators can be invoked for encoding and decoding (partial) data of 2D pictures carrying a grey shade function.
An exponential transform, the two dimensional analog of a similar operation on Cauchy integrals
introduced by A, Markov in his pioneering work on 1D moment problems, provides an effective dictionary
between “pictures” in the frequency domain and “matrices” in the state space interpretation.
A natural Riemann-Hilbert problem lies at the origin of this kernel with potential theoretic flavor. Quadrature domains for
analytic functions are singled out by a rationality property of the exponential transform, and hence an exact reconstruction
algorithm for this class of black and white shapes emerges. A two variable diagonal Pade approximation scheme and
some related complex orthogonal polynomials enter into the picture, with their elusive zero asymptotics.
Most of the results streaming from two decades of joint work with Bjorn Gustafsson.

This talk is part of the Isaac Newton Institute Seminar Series series.

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