Mar 6, 2014
In this paper we consider the equation ∇2φ + A(r2)X · ∇φ + C(r2)φ = 0 for X ∈ ℝN whose coefficients are entire functions of the variable r = |X|. Corresponding to a specified axially symmetric solution φ and set Cn of (n + 1) circles, an axially symmetric solution Λn*(x, η;Cn) and Λn(x, η;Cn) are found that interpolates to φ(x, η) on the Cn and converges uniformly to φ(x, η) on certain axially symmetric domains. The main results are the characterization of growth parameters order and type in terms of axially symmetric harmonic polynomial approximation errors and Lagrange polynomial interpolation errors using the method developed in [MARDEN, M.: Axisymmetric harmonic interpolation polynomials in ℝN, Trans. Amer. Math. Soc. 196 (1974), 385–402] and [MARDEN, M.: Value distribution of harmonic polynomials in several real variables, Trans. Amer. math. Soc. 159 (1971), 137–154].