Uniform approximation of functions
Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 421-456
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We obtain a criterion for the uniform approximability of functions by solutions of second-order homogeneous strongly elliptic
equations with constant complex coefficients on compact sets in $\mathbb{R}^2$ (the particular case of harmonic
approximations is not distinguished).
The criterion is stated in terms of the unique (scalar) Harvey–Polking capacity related to the leading coefficient of a
Laurent-type expansion (this capacity is trivial in the well-studied case of non-strongly elliptic equations).
The proof uses an improvement of Vitushkin's scheme, special geometric constructions, and methods of the theory of
singular integrals. In view of the inhomogeneity of the fundamental solutions of strongly elliptic operators
on $\mathbb{R}^2$, the problem considered is technically more difficult than the analogous problem
for $\mathbb{R}^d$, $d>2$.
Keywords:
uniform approximation, Vitushkin's scheme, capacities, homogeneous elliptic equations, Carleson measures.
@article{IM2_2021_85_3_a6,
author = {M. Ya. Mazalov},
title = {Uniform approximation of functions},
journal = {Izvestiya. Mathematics },
pages = {421--456},
publisher = {mathdoc},
volume = {85},
number = {3},
year = {2021},
language = {en},
url = {http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a6/}
}
M. Ya. Mazalov. Uniform approximation of functions. Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 421-456. http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a6/