To our knowledge the only result of this form is that of Askaripour and Barron. The literature has so far been largely concerned with uniform approximation, e.g., Gauthier and Sharifi for literature reviews see F. It should be observed that there are very few results involving approximation in the \(L^2\) norm by functions/forms on a larger region. The remainder of the paper is devoted to the proofs. 1.3, after the necessary preliminaries are dealt with in Sect. Precise statements of the theorems, as well as discussion of related literature, are given in Sect. These results are perhaps interesting on their own, and have several applications (see below).
Namely, we must extend the Schiffer isomorphism to this case, as well as the Plemelj–Sokhotski jump isomorphism and decomposition. To do so, we must first extend two of our results from the case of one curve separating the surface to many curves bounding conformal disks. We show that, given a compact Riemann surface R with nested regions \(\Sigma \subseteq \Sigma ' \subset R\) obtained by removing disks from R, under certain general conditions, the Dirichlet spaces of functions and Bergman spaces of one-forms on \(\Sigma '\) are dense in the Dirichlet spaces and Bergman spaces of \(\Sigma \), respectively.
In this paper, we apply these techniques to derive approximation theorems for nested multiply connected domains in Riemann surfaces of arbitrary genus. A number of results strongly indicate that quasicircles are the natural curves for this circle of ideas. Schiffer, which are also intimately related to approximations of holomorphic functions and forms, through the Faber and Grunsky operators and their generalizations. This theory involves certain singular integral operators due to M. The curves were assumed to be quasicircles, which are not rectifiable in general. In earlier publications, two of the authors developed a theory of the transmission of harmonic Dirichlet-bounded functions across curves, and of the related Plemelj–Sokhotski jump formula on such curves.
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