On the best polynomial approximation of generalized biaxisymmetric potentials in $$L^p$$-norm, $$p \geq 1$$.(English)Zbl 1273.30002

Summary: The real valued regular solution of generalized biaxially symmetric potential equation $\frac {\partial^2 F}{\partial x^2}+ \frac{\partial^2 F}{\partial y^2} + \frac {(2\alpha + 1)}{x} \frac{\partial F}{\partial x} + \frac{(2 \beta + 1)}{y} \frac{\partial F}{\partial y} =0, \quad \alpha > \beta > - \frac{1}{2}$ are called generalized biaxisymmetric potentials. In this paper, the characterization of lower order and lower type of entire GBASP $$F$$ in term of their approximation error $$E^p_n(F_\sigma)$$ in $$L^p$$-norm, $$p \geq 1$$ have been obtained. 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 one complex variable.

MSC:

 30B10 Power series (including lacunary series) in one complex variable 41A10 Approximation by polynomials 30E10 Approximation in the complex plane