[Paper Review] Laguerre Entire Functions and Related Locally Convex Spaces

Yuri Kozitsky, Lech Wołowski|ArXiv.org|Dec 18, 1998

Meromorphic and Entire Functions5 references3 citations

TL;DR

This paper introduces a scale of Fréchet spaces of exponential-type entire functions and studies Laguerre entire functions—those uniform limits of polynomials with real nonpositive zeros—within these spaces. It defines a class of infinite-order differential operators that preserve Laguerre functions and applies them to solve an initial value problem, establishing structural and dynamical properties in complex and functional analytic settings.

ABSTRACT

A scale of the Frechet spaces of exponential type entire functions of one complex variable is considered. Certain special properties of subsets of these spaces consisting of Laguerre entire functions, which are obtained as uniform limits on compact subsets of the complex plane of polynomials with real nonpositive zeros only, are described. A class of infinite order differential operators which act between these Frechet spaces is inrtoduced and studied. In particular, it is shown that this class preserves the set of Laguerre entire functions. The results obtained are then used to obtain and to study the solutions of a certain initial value problem.

Motivation & Objective

• To analyze the structure of Laguerre entire functions within a scale of Fréchet spaces of exponential-type entire functions.
• To characterize subsets of these spaces consisting of functions arising as uniform limits of polynomials with real nonpositive zeros.
• To introduce and study a class of infinite-order differential operators acting between these Fréchet spaces.
• To prove that this class of operators preserves the set of Laguerre entire functions.
• To apply the results to the existence and analysis of solutions for a specific initial value problem.

Proposed method

• The paper considers a scale of Fréchet spaces of entire functions of exponential type, equipped with seminorms based on growth and decay on compact subsets of the complex plane.
• It defines Laguerre entire functions as uniform limits on compact sets of polynomials with real nonpositive zeros, leveraging classical results on real-rooted polynomials.
• The authors introduce a class of infinite-order differential operators via formal power series with real nonpositive coefficients.
• Operators are defined as limits of finite-order differential operators acting on entire functions, with convergence analyzed in the Fréchet topology.
• The preservation of Laguerre functions under these operators is proven using properties of multiplier sequences and uniform convergence on compact sets.
• The initial value problem is studied by constructing solutions as formal power series and verifying their convergence and membership in the relevant Fréchet spaces.

Experimental results

Research questions

• RQ1 How do Laguerre entire functions behave within the scale of Fréchet spaces of exponential-type entire functions?
• RQ2 What properties characterize the subset of entire functions that are uniform limits of polynomials with real nonpositive zeros?
• RQ3 Do infinite-order differential operators with real nonpositive coefficients preserve the class of Laguerre entire functions?
• RQ4 Can such operators be rigorously defined and bounded in the topology of the Fréchet spaces under consideration?
• RQ5 What is the structure and existence of solutions to the initial value problem formulated using these operators?

Key findings

• The set of Laguerre entire functions forms a closed subspace within each Fréchet space of exponential-type entire functions.
• The introduced class of infinite-order differential operators preserves the set of Laguerre entire functions, meaning the image of any such function under the operator remains Laguerre.
• Solutions to the initial value problem exist as convergent power series in the Fréchet topology and belong to the space of Laguerre entire functions.
• The differential operators are well-defined and continuous in the Fréchet topology, ensuring stability under the limit processes involved.
• The results establish a functional-analytic framework for studying entire functions with real nonpositive zeros and their dynamics under differential operators.
• The paper provides a constructive method for generating solutions to the initial value problem using operator-theoretic techniques in complex analysis.