Growth and approximation of solutions to a class of certain linear partial differential equations in ℝN

Abstract

Sciences IntroductionThe work of Morris Marden ([13], [14]) focused on the study of polynomials, entire functions and their geometry in the complex plane. He utilized integral 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.

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

“…and P (α,β ) n (t) are Jacobi polynomials. Various authors such as Srivastava [3], McCoy [4], Kumar and Basu [5], Kumar and Bishnoi [6], Harfaoui [7], Kumar [8], Kadiri and Harfaoui [9], Kasana and Kumar [10]- [12] and Kapoor and Nautiyal [13] studied the growth and L p -approximation of regular real-valued solutions of certain elliptic partial differential equations but our results are different from these authors.…”

Criterion of the continuation of harmonic functions in the ball of n­dimensional space and representation of the generalized orders of the entire harmonic functions in ℝn in terms of approximation error

“…Paper [19] builts on the studies of [16]. Expressions for the order and type of solutions of certain linear differential equations in partial derivatives in terms of error in the axisymmetric harmonic polynomial approximation and the Lagrange interpolation were derived in [20]. In [21], a slow growth and the approximation of pseudoanalytic functions on disk were investigated.…”

Coefficients Characterization of Entire Harmonic Functions in Terms of Norm of Gradients at Origin in R^n, n≥ 3

“…In the study of entire functions of one complex variable, the main issues are the relationship between the growth of such functions and behavior of Taylor coefficients. Several authors such as Srivastava and Kumar [20], Kumar [11,14], Harfaoui [8] and others investigated growth parameters of entire functions in terms of Taylor's series coefficients and polynomial approximation errors in different norms. Similar studies have been done for harmonic functions by Kumar [12,13], Kumar and Kasana [15] and Armitage [1] as they have series expansion in terms of spherical harmonics in R n .…”