Bergman space

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In complex analysis, a branch of mathematics, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, <math>A^p(D)</math> is the space of holomorphic functions in D such that the p-norm

<math>|f|_p = left(int_D |f(x+iy)|^p,dx,dyright)^ < infty.</math>

Thus <math>A^p(D)</math> is the subspace of homolorphic functions that are in the space L<sup>p</sup>. The Bergman spaces are Banach spaces, which is a consequence of the estimate, valid on compact subsets K of D:Thus convergence of a sequence of holomorphic functions in L<sup>p</sup>(D) implies also compact convergence, and so the limit function is also holomorphic.

If p&nbsp;=&nbsp;2, then <math>A^p(D)</math> is a reproducing kernel Hilbert space, whose kernel is given by the Bergman kernel.


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