Vol. 81, No. 2, 1979

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Approximation and harmonic continuation of axially symmetric potentials in E3

Peter A. McCoy

Vol. 81 (1979), No. 2, 481–491
Abstract

Let F = F(x,y) be a complex valued axisymmetric potential (ASP) regular in the closed unit sphere about the origin in E3. Let the error in the approximation of F over n,ν (where n,ν is the set of all Newtonian potentials Rn.ν Pn(1∕Qν) generated from axisymmetric harmonic polynomials Pn and Qν by quasimultiplication) be defined by

En,ν(F) = inf{sup{|F (x,y) − Rn,ν(x,y)| : x2 + y2 = 1} : Rn,ν ∈ ℛn,ν}

n,ν = 0,1,2, . Then properties of the sequence {ρν}ν=0, ρν1 = limsupn→∞[En,ν(F)]1∕n, determine:

(i) the sphere to which F continues as an ASP with atmost (precisely) ν-singular circles (ii) the largest sphere of continuation as an ASP and (iii) the NASC for a sphere to contain infinitely many singular circles of the continuation of F.

Mathematical Subject Classification 2000
Primary: 31B15
Milestones
Received: 31 May 1978
Published: 1 April 1979
Authors
Peter A. McCoy