Abstract
For a given homogeneous elliptic partial differential operator \(L\) with constant complex coefficients, two Banach spaces \(V_1\) and \(V_2\) of distributions in \(\mathbb{R}^N\), and compact sets \(X_1\) and \(X_2\) in \(\mathbb{R}^N\), we study joint approximations in the norms of the spaces \(V_1 (X_1 )\) and \(V_2 (X_2 )\) (the spaces of Whitney jet-distributions) by the solutions of the equation \(L_u = 0\) in neighborhoods of the set \(X_1 \cup X_2\). We obtain a localization theorem, which, under certain conditions, allows one to reduce the above-cited approximation problem to the corresponding separate problems in each of the spaces.
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Vorontsov, A.M. Joint Approximations of Distributions in Banach Spaces. Mathematical Notes 73, 168–182 (2003). https://doi.org/10.1023/A:1022150807169
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DOI: https://doi.org/10.1023/A:1022150807169