Zeros of entire functions in strips and sectors

This work is a generalization of some results due to N. G. De Bruijn in three papers: Some theorems on the roots of polynomials (D1), An analogue of Graces Apolarity Theorem (D2), and The roots of trigonometric integrals (D3).; In chapter 1, two classes of entire functions are defined in Definitions 1 and 2. Using a classical result due to Korevaar (K), new characterizations are given for these classes in terms of their infinite-product representation in Corollaries 1 and 2. A classical result due to De Bruijn (D1) on the composition function of two polynomials is extended to the transcendental case in Proposition 2. A new algebraic characterization is given for these classes in Proposition 1. An analogue for the classical notion of multiplier sequence, due to Polya and Schur (PS) gives a new transcendental characterization of these classes in Proposition 3.; In chapter 2, the composition product is defined in Definition 9. This definition is a generalization of De Bruijn's (D2), by the incorporation of a parameter, and reduces to De Bruijn's definition for a specific value of this parameter. De Bruijn has shown that the Laguerre-Polya class is closed under the composition product. This result is generalized to a larger class of entire functions that includes the Laguerre-Polya class in Theorem 5. A known result on Fourier transforms, due to De Bruijn (D3), is generalized in Theorem 4 for this class. An analogue for the Hermite-Poulain theorem (O), is given in Theorem 3 for this class.