Amos Ron

The Interplay Between the Exponentials and the Polynomials in the Span of the Integer Translates of a Compactly Supported Function
or My Early Days in Madison

In 1987, Carl de Boor published a paper entitled "The polynomials in the span of the integer translates of a compactly supported function". In 1992, he published another paper, this time jointly with the speaker, entitled "The exponentials in the span of the integer translates of a compactly supported function". When I came to Madison, in 1988, I envisioned the latter paper to be a nice follow-up on the first. Instead, my discussions with Carl concerning the paper and the associated topic of local approximation by exponentials immediately ignited an evolution process that led to new basic understandings concerning the interplay between polynomials and exponentials. The resulted theory found applications in the areas of polynomials interpolation, polynomials ideals and their kernels, quasi-interpolation and more. And the aforementioned paper? More than two years passed before we found time to get back to it and and to finish it.
I will try to tell this story, focusing on the evolution, the motivation and the basic concepts, with less attention to the details and the fine results.