Approximation of entire function solutions of the Helmholtz equation having slow growth

Abstract

In this paper, we study the Chebyshev polynomial approximation of entire solutions of Helmholtz equations in R 2 in Banach spaces (B.p; q; m/ space, Hardy space and Bergman space). Some bounds on generalized order of entire solutions of Helmholtz equations of slow growth have been obtained in terms of the coefficients and approximation errors using function theoretic methods. X nD1 t 2n Q 2n .r 2 / is a real valued analytic function for t 2 OE 1; C1 that is entire for r 2 OE0; 1/ and known as Bergman "E function". The coefficients Q .2n/ are themselves entire Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/29/15 2:46 PM

Growth Estimates of Entire Function Solutions of Generalized Bi-Axially Symmetric Helmholtz Equation

“…Kumar [21] studied the growth and interpolation properties of solutions of above equation in several variables. Kumar [19,20] obtained some bounds on growth parameters of entire function solutions of Helmholtz equation in R 2 in terms of Chebyshev polynomial approximation errors in sup norm. Recently Kumar and Rajbir [23] considered the case µ = 0 and obtained the growth parameters such as lower order and lower type in terms of the coefficients in its Bessel-Gegenbauer series expansion.…”

Analog of the classical borel theorem for entire harmonic functions in ℝn and generalized orders

“…The order and type of potentials are expressed by the error of approximation. In [15], approximation of entire solutions of the Helmholtz equation by Chebyshev polynomials is studied, and some estimates are given on the growth parameters of these solutions in terms of coefficients and the approximation error with respect to the sup norm. In [16], research continues on the basis of [15].…”

“…In [15], approximation of entire solutions of the Helmholtz equation by Chebyshev polynomials is studied, and some estimates are given on the growth parameters of these solutions in terms of coefficients and the approximation error with respect to the sup norm. In [16], research continues on the basis of [15]. Paper [17] is devoted to the study of generalized and lower generalized q-types of solutions of the usual elliptic differential equation with partial derivatives.…”

Growth and L δ-approximation of solutions of the Helmholtz equation in a finite disk

“…Kumar [9,10] re ned the results of McCoy [17] and obtained some bounds on the growth parameters of entire function solutions of (1.1) in terms of coecients and Chebyshev approximation error in the sup norm. In a subsequent paper, Kumar [11] studied the Chebyshev polynomial approximation of entire solutions of the Helmholtz equation in ℝ 2 in Banach spaces. His results apply satisfactorily for slow growth.…”