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- Bergh, Jöran, 1941, et al.
(författare)
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Siciak's homogeneous extremal functions, holomorphic extension and a generalization of Helgason's support theorem
- 2019
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Ingår i: Annales Polonici Mathematici. - : Institute of Mathematics, Polish Academy of Sciences. - 0066-2216 .- 1730-6272. ; 123:1, s. 61-70
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Tidskriftsartikel (refereegranskat)abstract
- The main result of the present paper is that a function defined on a union of lines CE through the origin in C-n with directional vectors in E subset of C-n and holomorphic of fixed finite order and finite type along each of these lines can be extended to an entire function of the same order and finite type provided that CE is not pluripolar and all directional derivatives along the lines satisfy a necessary compatibility condition at the origin. We are able to estimate the indicator function of the extension in terms of Siciak's weighted homogeneous extremal function, where the weight is a function of the type of the given function on each given line. As an application we prove a generalization of Helgason's support theorem by showing how the support of a continuous function with rapid decrease at infinity can be located from partial information about the support of its Radon transform.
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| 2. |
- Berman, Robert, 1976
(författare)
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Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry
- 2019
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Ingår i: Annales Polonici Mathematici. - : Institute of Mathematics, Polish Academy of Sciences. - 0066-2216 .- 1730-6272. ; 123:1, s. 71-153
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Tidskriftsartikel (refereegranskat)abstract
- This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kahler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in C-n. A fair amount of background material is included. Along the way the results are generalized to the non-compact setting of C-n. This yields a probabilistic construction of Kahler solutions to Einstein's equations in C-n, with cosmological constant -beta, from a gas of interpolation nodes in equilibrium at positive inverse temperature beta. In the infinite temperature limit, beta -> 0, solutions to the Calabi-Yau equation are obtained. In the opposite zero temperature case the results may be interpreted as "transcendental" analogs of classical asymptotics for orthogonal polynomials, with the inverse temperature beta playing the role of the degree of a polynomial.
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