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Chui’s conjecture in Bergman spaces

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We solve an analog of Chui’s conjecture on the simplest fractions (i.e., sums of Cauchy kernels with unit coefficients) in weighted (Hilbert) Bergman spaces. Namely, for a wide class of weights, we prove that for every N, the simplest fractions with N poles on the unit circle have minimal norm if and only if the poles are equispaced on the circle. We find sharp asymptotics of these norms. Furthermore, we describe the closure of the simplest fractions in weighted Bergman spaces, using an \(L^2\) version of Thompson’s theorem on dominated approximation by simplest fractions.

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We are thankful to the referee for numerous interesting remarks and to Hervé Queffélec for helpful suggestions.

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Correspondence to Alexander Borichev.

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Communicated by Loukas Grafakos.

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This work was carried out in the framework of the project 19-11-00058 by the Russian Science Foundation.

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Abakumov, E., Borichev, A. & Fedorovskiy, K. Chui’s conjecture in Bergman spaces. Math. Ann. 379, 1507–1532 (2021).

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