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Approximation of entire harmonic functions in \(R^3\) in \(L^\beta\)-norm. (English) Zbl 1073.31002

Let \(H\) be harmonic on the open ball \(D_R\) of radius \(R\) in \(\mathbb{R}^3\) and continuous on \(\overline D_R\), and let \(\pi_n\) be the collection of all harmonic polynomials of degree at most \(n\). Further, let \(1\leq\beta\leq\infty\), let \(\|\cdot\|_{\beta,R}\) denote the \(L^\beta\) norm over \(D_R\), and let \(E^\beta_n(H, R)= \inf_{g\in\pi_n}\| H-g\|_{\beta,R}\). This paper presents necessary and sufficient conditions on the decay of \(E^\beta_n(H, R)\) as \(n\to\infty\) for H to have a harmonic continuation to all of \(\mathbb{R}^3\) with “\((p, q)\)-growth”.

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
41A10 Approximation by polynomials
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