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On approximation of generalized bi-axially symmetric potentials. (English) Zbl 0839.31009

Let \(\alpha, \beta > - 1/2\). A solution of the equation \[ \partial^2H/ \partial x^2 + \partial^2 H/ \partial y^2 + (2 \alpha + 1) y^{-1} \partial H/ \partial y + (2 \beta + 1) x^{-1} \partial H/ \partial x = 0 \] which is even in \(x\) and \(y\) is called a generalized bi-axially symmetric potential (GBSP). This paper presents growth and approximation results for such functions on the open disc \(D(R)\) of centre 0 and radius \(R\). The theorems are formulated in terms of generalized notions of order and type, which involve iterated logarithms of the maximum modulus of \(H\) over circles, and in terms of the approximation error \(E_n (H,r) = \inf_g |H - g |_r\), where \(r < R\) and \(|\cdot |_r\) denotes the supremum norm over \(\overline {D(r)}\) and where the infimum is taken over all GBSP polynomials \(g\) of degree at most \(2n\).

MSC:

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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