Measure, Integration & Real Analysis · Problem 8C.22

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · bergman-space-annulus

Problem

Suppose GG is the annulus defined by G={z∈C:1<∣z∣<2}.G = \{z \in \mathbf{C} : 1 < |z| < 2\}. (a) Find an orthonormal basis of La2(G)L^2_a(G). (b) Suppose f∈La2(G)f \in L^2_a(G) has Laurent series f(z)=∑k=−∞∞akzkf(z) = \sum_{k=-\infty}^{\infty} a_k z^k for z∈Gz \in G. Find a formula for ∥f∥\|f\| in terms of …,a−1,a0,a1,…\ldots, a_{-1}, a_0, a_1, \ldots.

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