Harmonic Approximation

Farrell sets for harmonic functions

“…It is interesting that the topological condition in Theorem 1 coincides with that obtained by Stray [14] in connection with Farrell sets for holomorphic functions, even though the two proofs have little in common and the theory of harmonic approximation differs significantly from its holomorphic counterpart (see [4]). This topological characterization also arises in connection with "Mergelyan sets" for holomorphic and harmonic functions (see [14] and [5]).…”

“…(We refer to Chapter 7 of [1] for an account of the notion of thinness.) Thus we can appeal to the Keldyš-Deny approximation theorem (Theorem 7.9.5 of [1], or Theorem 1.3 of [4]) to see that there is a harmonic function H on some neighbourhood of V such that…”

“…Since this latter set has a connected complement in R n , we can apply Walsh's theorem (Corollary 2.6.5 of [1], or Theorem 1.7 of [4]) to see that there is a harmonic polynomial q ε,r such that…”

Existence of Universal Taylor Series for Nonsimply Connected Domains

“…Theorems 1 and 2 will then be proved in the remaining two sections, along with their respective corollaries. The possibility that Theorem 2 might hold was initially suggested by earlier work of the author on the extension of superharmonic functions (see [8], or Chapter 6 of [9]). For the necessary potential theoretic background we refer to the books [22] and [1].…”

Non-tangential limits, fine limits and the Dirichlet integral

“…where (q k ) is an enumeration of Q, then u extends to a locally constant (and hence harmonic) function on an open set which contains E. It is easy to check that B * \E is connected and locally connected (see §3.2 of [9] for a discussion of local connectedness in this context). We note that, if z ∈ ∂B\A and 0 < δ < 1, then there exists a (smallest) number k 0 such that z ∈ F k0 and a number ε z,δ in (0, 1) such that…”