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

Authors : Devendra KUMAR

Pages : 156-162

Doi:10.33434/cams.439977

View : 11 | Download : 14

Publication Date : 2018-12-24

Article Type : Research Paper

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 insert ignore into journalissuearticles values(GBSP`s);. McCoy \cite{17} has showed that the rate at which approximation error $E^{\frac{p}{2n}}_{2n}insert ignore into journalissuearticles values(F;D); insert ignore into journalissuearticles values(p\ge 2,D$ 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 $E^{\frac{p}{2n}}_{2n}insert ignore into journalissuearticles values(F;D);$. 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