Measure, Integration & Real Analysis · Problem 8C.21

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · bergman-space-unit-disk

Problem

Let D\mathbf{D} denote the open unit disk in C\mathbf{C}; thus D={z∈C:∣z∣<1}.\mathbf{D} = \{z \in \mathbf{C} : |z| < 1\}. (a) Find an orthonormal basis of La2(D)L^2_a(\mathbf{D}). (b) Suppose f∈La2(D)f \in L^2_a(\mathbf{D}) has Taylor series f(z)=∑k=0∞akzkf(z) = \sum_{k=0}^{\infty} a_k z^k for z∈Dz \in \mathbf{D}. Find a formula for ∥f∥\|f\| in terms of a0,a1,a2,…a_0, a_1, a_2, \ldots. (c) Suppose w∈Dw \in \mathbf{D}. By the previous exercise and the Riesz Representation Theorem (8.47 and 8.76), there exists Γw∈La2(D)\Gamma_w \in L^2_a(\mathbf{D}) such that f(w)=⟨f,Γw⟩for all f∈La2(D).f(w) = \langle f, \Gamma_w\rangle \quad \text{for all } f \in L^2_a(\mathbf{D}). Find an explicit formula for Γw\Gamma_w.

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