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

Khan, Huzoor H.; Ali, Rifaqat

Transylv. J. Math. Mech. 3, No. 2, 103-110 (2011).

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

Keywords:

\(L^p\)-norm; lower order; lower type; best polynomial approximation error; entire function and generalized biaxisymmetric potentials