Results from the Theory of Entire Functions

Abstract

In this chapter, which consists of nine sections, we present elements from the theory of entire functions that are used throughout the book. The emphasis is to derive corollaries of the classical results of Phragmén-Lindelöf and Paley-Wiener that are used to derive completeness results for classes of operators. We fine-tune these results in various directions so that natural conditions in operator theory allow us to directly apply fundamental results from the theory of entire functions. In particular, we focus on the connection between the distribution of zeros and the growth properties of an entire function of completely regular growth. Such functions play an important role in the completeness results derived in this book. Using entire functions of the form

$$\displaystyle f(z) = p(z) + q(z) \int _{-a}^a e^{-zt} \varphi (t)\,dt,\qquad z \in {\mathbb C}, $$

where p and q ≠ 0 are polynomials, 0 < a < ∞, and φ is a non-zero square integrable function on the interval [−a, a], we show how to use the classical results from complex analysis to derive very detailed properties of these functions.