Geodesic
Criteria for $C^m$-approximability by bianalytic functions on planar compact sets
Sbornik. Mathematics, Tome 206 (2015) no. 2, pp. 242-281

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The paper puts forward criteria for approximability by bianalytic functions in the norms of the Whitney-type spaces $C^m$ on planar compact sets with $m \in (0, 2)$. These results, which are analogues of Vitushkin's well-known criteria for uniform rational approximation, together with results of O'Farrell and Verdera (the case $m \geqslant 2$) and Mazalov (the case $m=0$), provide a complete set of criteria for approximability by bianalytic functions for all $m \ge 0$. These conditions for approximability are obtained for both individual functions and (as corollaries) for classes of functions, using the terminology of geometric measure theory. Bibliography: 21 titles.
Keywords: $C^m$-approximation by bianalytic functions, bianalytic $C^m$-capacity, Hausdorff content of order $m$, Vitushkin-type localization operator. 
@article{SM_2015_206_2_a4,
     author = {M. Ya. Mazalov and P. V. Paramonov},
     title = {Criteria for $C^m$-approximability by bianalytic functions on planar compact sets},
     journal = {Sbornik. Mathematics},
     pages = {242--281},
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     number = {2},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2015_206_2_a4/}
}
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M. Ya. Mazalov; P. V. Paramonov. Criteria for $C^m$-approximability by bianalytic functions on planar compact sets. Sbornik. Mathematics, Tome 206 (2015) no. 2, pp. 242-281. http://geodesic.mathdoc.fr/item/SM_2015_206_2_a4/