Quasi topologies and rational approximation

Acharacterization of Newtonian functions with zero boundary values

“…In the Euclidean case with 1 < p < ∞ the previous result has been proved by Havin [17] and Bagby [4] for an open set E. Recently, the case p = 1 has been studied by Swanson in [31]. Our result holds true with an arbitrary set E, which is a slight generalization of known results already in the Euclidean case.…”

“…In the Euclidean setting this has been studied by Havin [17], Bagby [4], Swanson and Ziemer [32] and Swanson [31]. See also Theorem 9.1.3 in the monograph of Adams and Hedberg [1].…”

Two-dimensional magnetohydrodynamic equilibria with prescribed topology

“…Let ψ 2,i be a minimizer for given i, so ψ 2,i is harmonic in i and R 2 |∇ψ 2,i | 2 ≤ R 2 |∇ψ 1,i | 2 . Also, we can take ψ 2,i to be a quasi-continuous (see [4]) representative that equals ψ 1,i pointwise in R 2 \ i , i = 1, 2, 3. We claim for ψ 2 = 3 i=1 ψ 2,i that (ψ 2 , 0) ∈ D. Assuming this to be true, we note that R 2 |∇ψ 2,i | 2 < R 2 |∇ψ 1,i | 2 for some i = 1, 2, 3 would contradict the minimization property of (ψ 1 , 0) (recall that the support conditions im-…”

Approximation in the mean by solutions of elliptic equations

“…The original proofs of these results were constructive; proofs of the results of Runge and Mergeljan by the methods of functional analysis were obtained later [6;8,Chapter 2], but no such proof is known for the theorem of Vituskin. An analogue of the Vituskin problem for Lp approximation by holomorphic functions was considered by Havin [11], who used the methods of functional analysis and the Cartan fine topology, and by the author [3], who used the methods of functional analysis and quasitopologies. Hedberg [13] related these ideas to nonlinear potential theory, and obtained Wiener-type criteria for Lp approximation by holomorphic functions; further developments are given in the recent work of Hedberg and Wolff [18].…”

“…Hedberg [13] related these ideas to nonlinear potential theory, and obtained Wiener-type criteria for Lp approximation by holomorphic functions; further developments are given in the recent work of Hedberg and Wolff [18]. Lindberg [24,25] adapted the constructive techniques of Vituskin [36] to the study of L approximation by holomorphic functions, obtaining, in particular, a constructive proof of the approximation theorem of [3].…”