Rational approximation and weak analyticity. II
Let X be a compact subset of the complex plane \({\mathbb{C}}\), let \(0<\alpha <1\), and let \(R_{\alpha}(X)\) denote the closure in Lip(\(\alpha\),X) of the set of rational functions having poles off X. The first result is that a function \(f\in lip(\alpha,C)\) belongs to \(R_{\alpha}(X)\) if and only if there exists \(\eta(\delta)\downarrow 0\) as \(\delta\downarrow 0\), such that \[ | \int _{bdy B}f(\zeta)d\zeta | \leq \eta (side B)M^{1+\alpha}(B\sim X) \] for each closed square B. Here, \(M^{\beta}\) stands for \(\beta\)-dimensional Hausdorff content. This is a kind of quantitive Morera theorem. It resembles a theorem of Vitushkin on uniform approximation, but the method of proof is new. We define, for \(1<\beta <2\), the ``\(\beta\)-topology'' on \({\mathbb{C}}\). Theorem 2: Let \(0<\alpha <1\), \(\beta =1+\alpha\), \(p>2/(1-\alpha)\), and let f belong to the Sobolev space \(W^{1,p}_{loc}\). Then \(f\in R_{\alpha}(X)\) if and only if \(\frac{\partial f}{\partial \bar z}=0\) area-almost-everywhere on the \(\beta\)-interior of X. This is the Lip \(\alpha\) equivalent of a uniform approximation theorem from the preceding paper [Math. Ann. 273, 375-381 (1986; Zbl 0563.46022)].
theorem of Vitushkin on uniform approximation
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