A STUDY OF APPROXIMATION OF ENTIRE HARMONIC FUNCTIONS AND BORNOLOGICAL SPACES

Abstract

n 1978, Allan J. Fryant has studied the growth properties of entire harmonic functions. Later, he obtained the order and type of spherical harmonic functions in terms of the coefficients of their series expansions. T. B. Fugard extended these results for harmonic functions in . The approximation error of entire harmonic functions in ,R.3 was studied by G.P. Kapoor, A. Naitiyal, H.S. Kasana and others. The order and type of these functions were characterized in terms of their approximation errors. The present thesis presents a study of the growth of entire harmonic functions in Rn having index-pair (p,q) as well as the relation between the approximation error of entire harmonic functions and their growth. The various aspects of growth using proximate order and proximate type of entire harmonic functions in R3 of index-pair (p,q) have been studied. The study of spaces of entire functions was initiated by V.G. Iyer using the topological properties of the space. In 1971, H. Hogbe-Nlend introduced the concepts of bornology on a set. In 1981, M.D. Patwardhan extended this idea to spaces of entire functions. However, the space of entire functions represented by Drichlet series have not been studied so far and this has been investigated in details in the present thesis. The second part of the study pertains to the construction of the bornological spaces of entire functions of several variables as well as of analytic functions and studies the various properties of these spaces. The work embodied in the thesis can be divided mainly into five categories: . (i) Study of growth parameter of an entire harmonic function. (ii) Approximation error of an entire harmonic function. (iii) Approximation error of analytic functions. 20 (iv) Bornological properties of functions represented by power series. (v) Bornological properties of functions represented by Dirichlet series. Following is a brief account of the various topics covered in the thesis. The whole thesis has been divided into nine chapters. Chapter one: "Introduction". This chapter has two parts, part - A gives the literature review and part -B recapitulates the definitions and topics used in subsequent chapters. Only those topics that are relevant to the present work have been included. Chapter two: "Growth of entire harmonic functions in ,Rn , having index pair(p,q)". In this chapter expressions have been obtained for the (p, q) -order and (p, q) -type of non-constant entire harmonic functions in .R.n in terms of the norm of the m'th gradient of the harmonic function. In section one of this chapter, the concepts of the order and type to estimate the growth of a non-constant entire function has been, employed and the norm of the m'th gradient is defined by using the differential operator of the entire function. Section two presents some auxiliary results to help prove the main results. The expression for the (p, q) -order, (p, q) -type, lower (p, q) -order and lower (p,q)-type of entire harmonic functions in ,Rn in terms of the norm of the m'th gradient of the harmonic function is obtained in section three. Chapter three: "On the generalized type and approximation of an entire harmonic function in A3 with index pair (p,q)". Let JCR , 0 < R <00, be the class of all harmonic functions H in ,R3 , that are regular in the open ball DR of radius R centered at the origin and continuous on the closure DR of DR . For H E XR , set { En(H,R)= inf max _ II-1(x' , x2 , x3 ) — g(xi,x2,x3)1 , gE 17n (x1 ,x2,x3)E DR 21 where 17, consists of all harmonic polynomials of degree at most n . In this chapter the concept of ( p , q) - type of an entire harmonic function H in R3 with respect to the proximate order with index pair (p, q) has been introduced and its coefficient characterization in term of the approximation error E„ (H, R) is obtained. Chapter four: "On the generalized order and generalized type of entire harmonic functions". T.B. Fugard characterized the growth of entire harmonic functions h(x) in terms of the norms of m' th gradient V, h(0) . However his results leave a big class of functions for which the characterization cannot be obtained. In this chapter, use is made of generalized order introduced by M.N. Seremeta to characterize the growth of such entire harmonic functions. Results thus obtained generalize the results obtained by Fugard. Chapter five: "On the approximation of analytic functions of two complex variables by exponential polynomials". The order and type of the Dirichlet analytic function 00 f(si,s2) anin exp(siA„ + s2,un ) in terms of the rate of decay of the m,n=1 approximation error En„,(f, p) of the function f(si,s2) has been characterized by exponential polynomial. Class of all analytic functions f (s1, s2) is denoted by . In section one of this chapter, the concept of the growth of a function f (si,s2)E to characterize order p and type T of f (si, s2) E in terms of the rate of decay of the approximation error has been used. In section two some results have been obtained to provide the required proofs of our main results in section three. In section three, the abscissa of convergence of analytic function f in terms of the approximation error