Generalized growth and approximation errors of entire harmonic functions in \(R^n\), \(n \geq 3\)

Abstract

In this paper we study the continuation of harmonic functions in the ball to the entire harmonic functions in space \(\mathbb{R}^n\), \(n\geq 3\).
 The generalized order introduced by M.N. Seremeta has been used to characterize the growth of such functions. Moreover, the generalized order, generalized lower order and generalized type have been characterized in terms of harmonic polynomial approximation errors.
 Our results apply satisfactorily for slow growth.

GENERALIZED GROWTH AND APPROXIMATION OF HARMONIC FUNCTIONS IN ${R}^{n}, n\ge 3$

“…Several authors such as Srivastava and Kumar [12], Kumar (see [5], [8]), Harfaoui [2] 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 (see [6], [7]) and Kumar and Kasana [9] as they have series expansion in terms of spherical harmonics in R n .…”