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Linear independence of translations

Part of: Difference equations Difference and functional equations Harmonic analysis in one variable

Published online by Cambridge University Press:  09 April 2009

Joseph Rosenblatt
Joseph Rosenblatt
Affiliation:
Mathematics Department, Ohio State University, Columbus, OH 43210
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Abstract

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

MSC classification

Secondary: 39A10: Difference equations, additive 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 59 , Issue 1 , August 1995 , pp. 131 - 133
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Edgar, G. and Rosenblatt, J., ‘Difference equations over locally compact abelian groups’, Trans. Amer. Math. Soc. 253 (1979), 273–289.Google Scholar
[2]Guo, K., ‘On the p-approximate property for hypersurfaces of Rn’, Math. Proc. Cambridge Philos. Soc. 105 (1989), 503–511.Google Scholar
[3]Hömander, L., ‘Lower bounds at infinity for solutions of differential equations with constant coefficients’, Israel J. Math. 16 (1973), 103–119.Google Scholar