DOI: 10.2478/s12175-013-0192-4 Math. Slovaca 64 (2014), No. 1, 139–154 GROWTH AND APPROXIMATION OF SOLUTIONS TO A CLASS OF CERTAIN LINEAR PARTIAL DIFFERENTIAL EQUATIONS IN R N Devendra Kumar This work was done in the memory of Prof. H. S. Kasana, Senior Associate, ICTP, Trieste, Italy (Communicated by Giuseppe Di Fazio ) ABSTRACT. In this paper we consider the equation ∇ 2 ϕ + A(r 2 )X ·∇ϕ + C(r 2 )ϕ = 0 for X ∈ R N whose coefficients are entire functions of the variable r = |X|. Corresponding to a specified axially symmetric solution ϕ and set C n of (n + 1) circles, an axially symmetric solution Λ ∗ n (x, η; C n ) and Λ n (x, η; C n ) are found that interpolates to ϕ(x, η) on the C n 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 inter- polation polynomials in R N , 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]. c 2014 Mathematical Institute Slovak Academy of Sciences 1. Introduction The work of Morris Marden ([13], [14]) focused on the study of polynomi- als, entire functions and their geometry in the complex plane. He utilized in- tegral operators based on Laplace type integrals for Legendre polynomials to associate harmonic functions with analytic functions of single complex variable 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 41A05; Secondary 31B05, 31B10. K e y w o r d s: axisymmetric harmonic polynomials, axi-convex region, order and type, Lagrange polynomial, approximation and interpolation errors, Bergman operator and hypersphere.