CLOSURE AND EXPANSIONS IN SERIES OF COMPLEX EXPONENTIALS

Let (LAMDA) be a set of complex numbers and let E(,(LAMDA)) be the set of exponential functions {lcub}e('i(lamda)t){rcub}(,(lamda) (ELEM) (LAMDA)) where t is one real or complex variable. We will say that E(,(LAMDA)) is closed in a topological vector space of complex-valued functions if the subspace spanned by E(,(LAMDA)) is dense in the space. In that case, each function in the space is a limit of finite linear combinations of functions in E(,(LAMDA)) and may consequently be written as an infinite series of finite sums of exponentials from that set; the series converges to the function in the particular topology of the space. In some instances, it is possible to find special Fourier-type expansions of functions in which the series coefficients may be prescribed and each exponential function appears in at most one term of the series.; This expository work provides an introduction to some aspects of these closure and expansion problems in the three familiar function spaces C{lcub}a,b{rcub}, L('p){lcub}a,b{rcub} and H(D), where H(D) is the space of all functions analytic on a simply connected domain D of the complex plane with the topology of uniform convergence on compact subsets of D.; In each of the spaces the Hahn-Banach theorem is valid and the condition for closure can be restated in terms of the set of zeros of certain entire functions of exponential type. Chapter I provides some background material concerning entire functions and a discussion of the three spaces and their dual spaces. Chapter II contains historical background and the basic results about the closure of; (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI); in L('p){lcub}a,b{rcub} and C{lcub}a,b{rcub} and about the related Fourier series. The closure of more general sets E(,(LAMDA)) in C{lcub}a,b{rcub} and L('p){lcub}a,b{rcub} is investigated in Chapter III, and nonharmonic Fourier expansions in L('p){lcub}a,b{rcub} are studied in Chapter IV. Chapter V is devoted to closure and expansion problems in H(D).