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 .