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
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.
The paper deals with the characterization of generalized order and generalized type of entire functions in several complex variables in terms of the coefficients of the development with respect to the sequence of extremal polynomials and the best L p -approximation and interpolation errors, 0 < p ≤ ∞, on a compact set K with respect to the setwhere V K is the Siciak extremal function of a L-regular compact set K or V K is the pluricomplex Green function with a pole at infinity. It has been noticed that in the study of growth of entire functions, the set K r has not been used so extensively in comparison to disk. Our results apply satisfactorily for slow growth in C n , replacing the circle {z ∈ C; |z| = r } by the set K r and improve and extend various results of
Abstract. The (p, q)-growth of entire function solutions of Helmholtz equations in R 2 has been studied. We obtain some lower bounds on order and type through function theoretic formulae related to those of associate. Our results extends and improve the results studied by McCoy [10].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.