Best ${L^p}$-approximation of generalized biaxially symmetric potentials over Carath'eodory domains in ${C^N}$ having slow growth

Abstract

Let $ F$ be a real valued generalized biaxially symmetric potentials (GBASP) defined on the Caratheodory domain on $ C^N$. Let $ L_\mu^p(D)$ be the class of all functions $ F$ holomorphic on $ D$ such that $ \parallel F\parallel_{D,p}=[\int_D\mid F\mid^pd\mu]^{1\over p}$. Where $ \mu$ is the positive finite, Boral measure with regular asymptotic distribution on $ C^N$. For $ F\in L_{\mu}^p(D)$, set $ E_n^p(F)=\inf\{\parallel F-P\parallel_{D,p}:P\in H_n\}$, $ H_n$ consist of all real biaxisymmetric harmonic polynomials of degree at most $ 2n$. The paper deals with the growth of entire function GBASP in terms of approximation error in $ L_{\mu}^p$-norm on $ D$. The analysis utilizes the Bergman and Gilbert integral operator method to extend results from classical function theory on the best polynomial approximation of analytic functions of several complex variables. Finally we prove a generalized decomposition theorem in a new way. The paper is the generalization of the concepts of generalized growth parameters to entire functions on Caratheodory domains on $ C^N$ (instead of entire holomorphic functions on $ C$) for slow growth.