Entire families of functions which are canonical products

Abstract

The object of this thesis is to extend certain definitions and theorems well known in the theory of entire functions in order to establish analogous results for entire families of functions. This will make it possible to pursue further certain theorems on families of functions which have gaps in their coefficient sequences such that the circle of regularity is a singular line (cut) for the family. These considerations follow along the lines investigated and suggested by Fabry, Faber, Borel, Hadamard, Mandelbrojt, Johnson, and others. The definitions of order, zeros, exponent of convergence, and genus for families suggested themselves by the necessities of the proofs of subsequent theorems when extended to families of functions. In his thesis Johnson [3, p. 34] describes conditions permitting regular continuation of families of functions. In the course of this work, he defines an entire family of functions, the family maximum modulus, and the family exponential type. These ideas will be extended and certain similarities will be emphasized. The description of the set of zeros of a function by means of a growth sequence will be defined for a family of functions. The growth sequence of a family will be related to the family order as it is done for a single function by means of the exponent of convergence of its zeros. These results seem to indicate that it might be possible to extend a gap theorem of Mandelbrojt for single functions [9, p. 344] to the case of a family of functions such that the domain of regularity of one family is described in terms of the growth sequence of another family. The material in this thesis is intended to be partly expository and partly investigative in nature. The author expresses his sincere appreciation and gratitude to Professor Guy Johnson whose active encouragement and interest have made the writing of this thesis an inspiration and a pleasure.