ABSTRACT
We now go back to studying the Szego˝ kernel function on a general bounded domain Ω with C∞ smooth boundary. In particular, Ω is again allowed to be multiply connected.
For fixed a ∈ Ω, we will let Sa(z) denote the function of z given by Sa(z) = S(z, a). Let Σ denote the (complex) linear span of the set of functions {Sa(z) : a ∈ Ω}. It is easy to see that Σ is a dense subspace of H2(bΩ). Indeed, if h ∈ H2(bΩ) is orthogonal to Σ, then h(a) = 〈h, Sa〉b = 0 for each a ∈ Ω; thus h ≡ 0. In this chapter, we will prove that Theorem 9.1. Σ is dense in A∞(Ω).
