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Published online by Cambridge University Press: 25 June 2025
The recent developments in the function theory of the Bergman space are reviewed. Key ingredients are: factorization based on extremal divisors, an analog of Beurling's invariant subspace theorem, concrete examples of invariant subspaces of index higher than one, a partial description of zero sequences, characterizations of interpolating and sampling sequences, and some remarks on weighted Bergman spaces.
1. The Hardy and Bergman Spaces: A Comparison
The Hardy space H2 consists of all holomorphic functions on the open unit disk 𝔻 such that. where 𝕋 stands for the unit circle and ds is arc length measure, normalized so that the mass of 𝕋 equals 1. In terms of Taylor coefficients, the norm takes a more appealing form. The Bergman space, on the other hand, consists of all holomorphic functions on 𝔻 where dS is area measure, normalized so that the mass of 𝔻 equals 1.
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