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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
MathSciNet review: 3020848
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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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  • 1. S. Bochner, Boundary values of analytic functions in several variables and of almost periodic functions, Ann. of Math. (2) 45 (1944), 708–722. MR 0011132
  • 2. C. Caratheodory, Theory of functions of a complex variable. Vol. 2, Chelsea Publishing Company, New York, 1954. Translated by F. Steinhardt. MR 0064861
  • 3. Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
  • 4. Henry Helson and David Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165–202. MR 0097688
  • 5. Henry Helson and David Lowdenslager, Prediction theory and Fourier series in several variables. II, Acta Math. 106 (1961), 175–213. MR 0176287
  • 6. Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
  • 7. A tribute to Henry Helson, Notices Amer. Math. Soc. 58 (2011), no. 2, 274–288. Donald Sarason, coordinating editor. MR 2768120
  • 8. Victor L. Shapiro, Bounded generalized analytic functions on the torus, Pacific J. Math. 14 (1964), 1413–1422. MR 0172072
  • 9. A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776

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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: https://doi.org/10.1090/S0002-9939-2012-11442-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.