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\(L^p\)-approximation of generalized biaxially symmetric potentials over Carathéodory domains. (English) Zbl 1150.41357

Summary: Let \(F^{\alpha ,\beta }\) be a real generalized biaxially symmetric potentials (GBASP) defined on the Carathéodory domain and let \(L^\rho (D)\) be the class of functions \(F^{\alpha ,\beta }\) holomorphic in \(D\) such that \(\| F^{\alpha ,\beta }\| _{D,p} =\left (A^{-1}\iint \limits _D| F^{\alpha ,\beta }| \operatorname {d}\!x\,{\operatorname {d}}y\right )^{1/p}\), \(A\) is the area of the domain \(D\). For \(F^{\alpha ,\beta }\in L^p(D)\), set \(E^p_n\bigl (F^{\alpha ,\beta }\bigr ) ={\text{inf}}\{\| F^{\alpha ,\beta } - P^{\alpha ,\beta }\| _{D,p} \:P^{\alpha ,\beta }\in H_n\}\), \(H_n\) consists of all even biaxially symmetric harmonic polynomials of degree at most 2\(n\). This paper deals with the growth of entire function GBASP in term of approximation error in \(L^p\)-norm on \(D\). The analysis utilizes the Bergram and Gilbert integral operator method to extend result from clasical function theory on the best polynomial approximation of analytic functions of a complex variable.

MSC:

41A80 Remainders in approximation formulas
35J15 Second-order elliptic equations
30E10 Approximation in the complex plane
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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