Reconstruction of Planar Domains from Partial Integral Measurements

Dmitry Batenkov, Vladimir Golubyatnikov, Yosef Yomdin

Research output: Contribution to journalMeeting Abstractpeer-review


We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewise-algebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on the one-dimensional reconstruction method of [5] and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given. Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects.
Original languageEnglish
Pages (from-to)51-66
Number of pages16
JournalComplex Analysis And Dynamical Systems V
StatePublished - 2013
Event5th International Conference on Complex Analysis and Dynamical Systems (CA&DS V) - Akko, ISRAEL
Duration: 22 May 201127 May 2011


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