Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions

On radial limit functions for entire solutions of second order elliptic equations in $\mathbf{R}^2$

“…This space satisfies the conditions (1)- (4) of [2]. From the fact that V is locally equivalent to the space C 1 (R 2 ) and from the approximation properties of f on F R 0 mentioned above, it follows also that there exists a locally finite family of balls covering F 0 such that for each ball B in this family and for each ε > 0, there exists g such that Lg = 0 on some neighbourhood of F 0 ∩ B and f − g F0∩B < ε i.e.…”

“…f is approximable locally on F 0 in the norm of V by (local) Lanalytic functions. Theorem 2 in [2] now states that this is equivalent to global approximation, that is, for each ε > 0, there exists an L-analytic…”

“…Since R 2 ∞ \ F 0 is connected and locally connected (that is, F 0 is a RKL-set in the terminology of [2] (the letters stand for Roth-KeldyshLavrentieff)), we can use an analog of Runge's theorem obtained in [2, Theorem 1] to approximate in the norm of V L-analytic functions on F 0 by L-entire functions. We thus conclude that we can find an L-entire function h such that f − h F0 ≤ 1.…”

Maximal Cluster Sets of L-Analytic Functions Along Arbitrary Curves

“…We have that 1 ∈ ran C ϕ for any holomorphic selfmapping ϕ. 4. We know that internal control implies continuity.…”

“…Following [4], a relatively closed subset F in Ω will be called a RothKeldysh-Lavrentiev set, or more simply an Ω-RKL set, if Ω * \ F is connected and locally connected. The following lemma (see [3] and [6]) will reveal useful in the proof of our main result.…”