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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Generalized analyticity on the $ N$-torus


Author: Victor L. Shapiro
Journal: Proc. Amer. Math. Soc. 141 (2013), 1605-1612
MSC (2010): Primary 42A16; Secondary 42A63
Published electronically: September 26, 2012
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Abstract: In dimension $ N=2,$ let $ v$ be a unit vector of irrational slope. With $ T_{2} $ the 2-torus, say $ f\in \mathcal {A}_{v}^{+}$ if $ f\in L^{1}(\mathbf {T}_{2})$ and $ m\cdot v<0$ $ \Rightarrow \widehat {f}\left (m\right ) =0$ for $ m$ an integral lattice point. Let $ G_{v}$ be the one-parameter group generated by $ v$ on $ T_{2},$ and let $ E\subset G_{v}$ be a closed and bounded set. Call $ E$ a set of uniqueness for $ \mathcal {A}_{v}^{+}\cap C\left (T_{2}\right ) $ if $ f\in \mathcal {A}_{v}^{+}\cap C\left (T_{2}\right ) $ and if $ \ f\left (x\right ) =0$ for $ x\in E$ implies $ f\equiv 0$ on $ T_{2}.$ The following result is established$ :$ A necessary and sufficient condition that E be a set of uniqueness for $ \mathcal {A} _{v}^{+}\cap C\left (T_{2}\right ) $ is that E is a set of positive linear measure.


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Additional Information

Victor L. Shapiro
Affiliation: Department of Mathematics, University of California, Riverside, California 92521
Email: shapiro@math.ucr.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11442-X
PII: S 0002-9939(2012)11442-X
Keywords: Fourier coefficient, one-parameter group, positive linear measure
Received by editor(s): March 15, 2011
Received by editor(s) in revised form: August 26, 2011
Published electronically: September 26, 2012
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.