On uniform approximation by harmonic functions (Q1099350)

Extending author's previous work on connections between approximation by analytic functions and the isoperimetric inequality [J. Funct. Anal. 58, 175-193 (1984; Zbl 0551.46032), J. Approximation Theory 50, 127-132 (1987; Zbl 0612.41024), Proc. Am. Math. Soc. 101, 475-483 (1987)] the concept of the harmonic content \(\Delta\) (\({\mathfrak X})=^{def}dist_{C({\mathfrak X})}(| X|\) 2,H(\({\mathfrak X}))\) is introduced in the paper. Here, \({\mathfrak X}\subset {\mathbb{R}}^ n \)is a compact set, \(| X|\) \(2=\sum^{n}_{1}X\) \(2_ i\), \(X=(X_ 1,...,X_ n)\in {\mathbb{R}}^ n \)and H(\({\mathfrak X})=^{def}\) closure of functions harmonic in a neighborhood of \({\mathfrak X}\) in the space C(X) of all continuous functions on \({\mathfrak X}\). It is shown that \(\Lambda (X)=0\Leftrightarrow\) H(\({\mathfrak X})=C({\mathfrak X})\) (the ``Stone- Weierstrass theorem for H(\({\mathfrak X})'')\). Also, \(\Lambda\) (\({\mathfrak X})\) is shown to enjoy a simple geometric estimate (\(\Lambda\) (\({\mathfrak X})\leq (1/2)R^ 2_{{\mathfrak X}}\), \(R_{{\mathfrak X}}\) is the ``volume radius'' of \({\mathfrak X}\) which is sharp (it becomes equality for balls). After this paper had been published, it was pointed out that the main theorem can also be derived from the results of \textit{W. Hansen} in fine potential theory [Ill. J. Math. 29, 103-107 (1985)] obtained, however, by completely different methods. Recently, the author generalized the above results to quite a general class of uniformly elliptic operators and also, obtained the sharp lower bounds for \(\Lambda\) (\({\mathfrak X}):\Lambda\) (\({\mathfrak X})\geq (1/2)\tilde R^ 2_ X\), where \(\tilde R_ X\) is the ``harmonic radius'' of \({\mathfrak X}\).