On Growth and Approximation of Generalized Biaxially Symmetric Potentials on Parabolic-Convex Sets

Abstract

The regular, real-valued solutions of the second-order elliptic partial differential equation\beq \frac{\prt^2F}{\prt x^2} + \frac{\prt^2F}{\prt y^2} + \frac{2\alpha+1}{x} \frac{\prt F}{\prt y} + \frac{2\beta+1}{y} \frac{\prt F}{\prt x} =0, \alpha,\beta>\frac{-1}{2}\eeq are known as generalized bi-axially symmetric potentials (GBSP's). McCoy \cite{17} has showed that the rate at which approximation error $E^{\frac{p}{2n}}_{2n}(F;D),(p\ge 2,D$ is parabolic-convex set) tends to zero depends on the order of $GBSP$ F and obtained a formula for finite order. If $GBSP$ F is an entire function of infinite order then above formula fails to give satisfactory information about the rate of decrease of $E^{\frac{p}{2n}}_{2n}(F;D)$. The purpose of the present work is to refine above result by using the concept of index-q. Also, the formula corresponding to $q$-order does not always hold for lower $q$-order. Therefore we have proved a result for lower $q$-order also, which have not been studied so far.

Keywords

• parabolic-convex set
• index-q
• q-order
• lower q-order
• generalized bi-axially symmetric potentials and elliptic partial differential equation