*Selected papers of Norman Levinson* Preface

In Levinson's biography we quote the first paragraph of the Preface of these volumes. Here we gave an even fuller quote from the Preface:- |

The deep and original ideas of Norman Levinson have had a lasting impact on fields as diverse as differential and integral equations, harmonic, complex and stochastic analysis, and analytic number theory during more than half a century. Yet, the extent of his contributions has not always been fully recognized in the mathematics community. For example, the horseshoe mapping constructed by Stephen Smale in 1960 played a central role in the development of the modern theory of dynamical systems and chaos. The horseshoe map was directly stimulated by Levinson's research on forced periodic oscillations of the Van der Pol oscillator, and specifically by his seminal work initiated by Cartwright and Littlewood. In other topics, Levinson provided the foundation for a rigorous theory of singularly perturbed differential equations. He also made fundamental contributions to inverse scattering theory by showing the connection between scattering data and spectral data, thus relating the famous Gelfand- Levitan method to the inverse scattering problem for the Schrödinger equation. He was the first to analyze and make explicit use of wave functions, now widely known as the Jost functions. Near the end of his life, Levinson returned to research in analytic number theory and made profound progress on the resolution of the Riemann hypothesis.

Levinson's papers are typically tightly crafted and masterpieces of brevity and clarity. It is our hope that the publication of these selected papers will bring his mathematical ideas to the attention of the larger mathematical community.

In these two volumes, Levinson's papers are grouped by themes, rather than chronologically. Approximately one half of Levinson's work was devoted to differential and integral equations, and Volume 1 is devoted to papers in these areas arranged by topics in seven chapters. The complete lists of his publications and of his 34 Ph.D. students begin on p. xxvii of Volume 1. Volume 2 presents papers in harmonic, complex and stochastic analysis, and in number theory arranged in four major themes. Finally, Chapter XII of Volume 2 is devoted to papers on miscellaneous topics that point to the enormous breadth of Levinson's interests. Commentaries on most of Levinson's principal contributions by researchers active in each topic introduce the twelve chapters.

Although not reproduced in these volumes, two of Levinson's four books have become classics: The AMS Colloquium Publication *Gap and density theorems,* Amer. Math. Soc., New York, 1940, originally published in 1940 and still in print, subsumes much of Levinson's brilliant early research in harmonic and complex analysis. The essay by Raymond Redheffer on the continuing impact of this work on current research appears in Chapter VIII. The advanced text on ordinary differential equations *Theory of ordinary differential equations,* McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955, co-authored with Earl Coddington has literally become the bible for students that helped train several generations of mathematicians, scientists and engineers since it was published in 1955.

JOC/EFR March 2006

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