Abstract
In 1967, I sent some reprints to the Institute for Advanced Study as part of an application for a visiting membership. In one of these papers [5], Arne Beurling saw a result of mine which suggested to him a conjecture which he subsequently proved. The result is still scarcely known. It is a real variable result whose proof uses some complex analysis and harmonic analysis. It has implications for the study of one-parameter semigroups of operators, Markov processes, structure of Banach spaces, and certain aspects of chaos theory. It is the intent of this article to describe the result, give the setting under which it was found, and indicate some of its implications. I also hope to contribute to a fuller picture of Beufling. First, I describe what Beurling saw which led to the discovery of his theorem. We need some notation for higher-order finite differences: There is the following easy consequence:
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