Global ( p , q )-Growth of Entire Harmonic Functions in R n in Terms of Approximation Errors

The relationship between the generalized growth parameters of an entire harmonic function in space R, n ≥ 3, with the rate of its best harmonic polynomial approximation error and ratios of these errors of functions harmonic in the ball of radius R has been studied. 2010 Mathematics Subject Classifications: 30E10, 41A15


Introduction
The approximation of entire functions on compact sets was studied by Srivastava and Kumar [14,15] and obtained generalized growth parameters in terms of approximation and interpolation error.Similar studies have been done for harmonic functions.The harmonic functions play an important role not only in theoretical mathematics but also in Physics and mechanics to describe different stationary processes.Therefore, it is significant to mention here that the study of generalized growth parameters of a harmonic function in an n-dimensional spaces has relevance.Harmonic functions can be expanded into series in spherical harmonics in space R n , n ≥ 3 and in the adjoined Legendre polynomials in space R 3 .The growth characteristics of harmonic functions in terms of the coefficients of their expansion into series as well as not related the expansion coefficients, in particular, in terms of the norm of their gradient at the origin were obtained.Also, the growth of harmonic function in terms of approximation errors by harmonic polynomials in R n , n ≥ 3 was considered by various authors (see, [3,[6][7][8][9][10][11][12][13]).The aim of the present work is to investigate conditions under which a harmonic function in the ball of n-dimensional space continues to the entire harmonic function, and to derive formulae for the generalized growth parameters (p, q)-order, lower(p, q)-order, (p, q)-type and lower(p, q)-type of harmonic function in space in terms of harmonic polynomial approximation errors.Here p and q are integers such that p ≥ q ≥ 1.Let u be an entire harmonic function in R n and has a Fourier-Laplace series expansion [16] where x ∈ S n = {x ∈ R n : |x| = 1} a unit sphere in R n centered at the origin Here dS is the element of the surface area on the sphere S n , (u, We denote H R , the class of harmonic functions in B n R and continuous on B n R , 0 < R < ∞.Let π k be the set of harmonic polynomials of degree≤ k.The approximation error of function u ∈ H R by harmonic polynomials P ∈ π k be defined as For u ∈ H R continue to the entire harmonic function of n-dimensional space R n , n ≥ 3, it is known [17 ,p.45] that lim 3) The concept of order ρ(F ) and lower order λ(F ) of an entire function F (z) = ∞ n=0 a n z n was introduced by R.P. Boas [1] as The concept of type T (F ) and lower type t(F ) has been introduced when the entire functions have same nonzero finite order.An entire function of order ρ, 0 < ρ < ∞, is said to be of type T (F ) and lower type t(F ) if For the class of order ρ(F ) = 0 and ρ(F ) = ∞, the type can not be defined.To refine the above concept of order and type, Juneja et.al., [4,5] introduced the concept of (p, q)orders and (p, q)-types.Therefore, we define the (p, q)-order and lower(p, q)-order as The (p, q)-type and lower(p, q)-type are defined as where log Notations: From [4] we define the relations between(p, q)-order, lower(p, q)-order, the coefficients of F (z) and ratios of these successive coefficients as following: n=0 a n z n be an entire function of (p, q)-order ρ(p, q, F ), then where a n z n be an entire function of (p, q)-order ρ(p, q, F ), then where a n z n be an entire function of (p, q)-order ρ(p, q, F ) and (| an a n+1 |) a nondecreasing function of n for n > n 0 then λ(p, q, F ) = P (l(p, q, F )) where a n z n be an entire function of (p, q)-order ρ(p, q, F ) and (| an a n+1 |) a nondecreasing function of n for n > n 0 then λ(p, q, F ) = P (l * (p, q, F )) where .
From [5] we define the relation between (p, q)-type, lower(p, q)-type and the coefficients of F (z) as: a n z n be an entire function of (p, q)-order ρ(p, q, F ) and (p, q)-type T (p, q, F ) if and only if T = M V , where a n z n be an entire function of (p, q)-order ρ(p, q, F ), lower(p, q)-type t(p, q, F ) and (| an a n+1 |) a nondecreasing function of n for n > n 0 then t = M v, where

Auxiliary Results
In this section we will prove some auxiliary results which will be used in the sequel.Consider the two functions f and g of complex variable z: and In view of [17, pp. 47] we see that if u is an entire function then f and g are also entire functions of the complex variable z.Using Lemma 3 with inequality (8) of [17], we get where m(r, f ) is the maximum term of power series of function f (z) on the circle{z : |z| = r}, and M (r, g) = max |z|=r |g(z)|.Lemma 2.1.Let f and g be defined by (2.1) and (2.2).Then the (p, q)-orders and (p, q)-types of f and g respectively are equal.
Proof.First we consider the case (p, q) = (2, 1), ) Using Theorem A, we see that the function f and g have same (p, q)-order, it leads to the fact that ρ(p, q, f ) = ρ(p, q, g) = ρ.Now we consider the (p, q)-type for q = 2 as 2] k .
Similarly for g we have 2] k .
Now for the case q ≥ 3, we have 2] k .