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# A convergent series

Let be holomorphic on the open unit disk and suppose that the integral converges. If the Taylor expansion of is of the form then prove that the sum

converges.

Solution

We evaluate the integral using the standard orthogonality results for . Thus:

and thus the series converges.

Note: The set of functions satisfying this is a Hilbert space of functions, but it is not the same as the Hardy space .