Measure, Integration & Real Analysis · Problem 8C.24

Real and Complex Measures — Sheldon Axler

Mathematics — Measure Theory · dirichlet-space

Problem

The Dirichlet space D\mathcal{D} is defined to be the set of analytic functions f:D→Cf: \mathbf{D} \to \mathbf{C} such that ∫D∣f′∣2 dλ2<∞.\int_{\mathbf{D}} |f'|^2\, d\lambda_2 < \infty. For f,g∈Df, g \in \mathcal{D}, define ⟨f,g⟩\langle f, g\rangle to be f(0)g(0)‾+∫Df′g′‾ dλ2f(0)\overline{g(0)} + \int_{\mathbf{D}} f'\overline{g'}\, d\lambda_2. (a) Show that D\mathcal{D} is a Hilbert space. (b) Show that if w∈Dw \in \mathbf{D}, then f↦f(w)f \mapsto f(w) is a bounded linear functional on D\mathcal{D}. (c) Find an orthonormal basis of D\mathcal{D}. (d) Suppose f∈Df \in \mathcal{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. (e) Suppose w∈Dw \in \mathbf{D}. Find an explicit formula for Γw∈D\Gamma_w \in \mathcal{D} such that f(w)=⟨f,Γw⟩for all f∈D.f(w) = \langle f, \Gamma_w\rangle \quad \text{for all } f \in \mathcal{D}.

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