Generalized analyticity on the $N$-torus
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- by Victor L. Shapiro PDF
- Proc. Amer. Math. Soc. 141 (2013), 1605-1612 Request permission
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.References
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Additional Information
- Victor L. Shapiro
- Affiliation: Department of Mathematics, University of California, Riverside, California 92521
- Email: shapiro@math.ucr.edu
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1605-1612
- MSC (2010): Primary 42A16; Secondary 42A63
- DOI: https://doi.org/10.1090/S0002-9939-2012-11442-X
- MathSciNet review: 3020848