J Austral Math Soc Ser A 59 pp131--133, 1995.

Linear Independance of Translations

Joseph Rosenblatt

Abstract

It was shown by Edgar and Rosenblatt that if f Î L_p(Rⁿ), 1 £ p < 2n/(n - 1), and f ¹ 0, then f has linearly independant translates. Using a result of Hormander, it is shown here that the same theorem holds if p = 2n/(n - 1). This gives a sharp result because for n ³ 2, there exists f Î C₀(Rⁿ), f ¹ 0, which is simultaneously in all L_p(Rⁿ), p > 2n/(n - 1), that has a linear dependence relation among its translates. References and some discussion are included.

1991 AMS Subject Classification: 39A10, 42A38

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Authors

Joseph Rosenblatt

Mathematics Department, Ohio State University, Columbus, OH 43210.

Current Address:

Mathematics Department, University of Illinois, Urbana, IL 61801.