Some properties of measures with discrete support

S. Yu. Favorov

doi:10.15330/ms.46.2.189-195

We give some new conditions for the support of a discrete measure on Euclidean space to be a finite union of translated lattices. In particular, we consider the case when values of masses $a_{\lambda}$ of discrete measure satisfy the equality $G(a_{\lambda},\bar a_{\lambda})=0$ for each analytic function $G(z,w)$.

1. A. Cordoba, Dirac combs, Lett. Math. Phis., 17 (1989), 191–196.

2. P.J. Cohen, On a conjecture of Littlewood and idempotent meagures, Am. J. Math., 82 (1960), 191–212.

3. S.Yu. Favorov, Bohr and Besicovitch almost periodic discrete sets and quasicrystals, Proc. Amer. Math. Soc., 140 (2012), 1761–1767.

4. S.Yu. Favorov, Fourier quasicrystals and Lagarias’ conjecture, Proc. Amer. Math. Soc., 144 (2016), 3527–3536.

5. M.N. Kolountzakis, On the structure of multiple translations tilings by polygonal regions, Preprint, 1999, 16 p.

6. M.N. Kolountzakis, J.C. Lagarias, Structure of tilings of the line by a function, Duke Math. Journal, 82 (1996), 653–678.

7. J.C. Lagarias, Mathematical quasicrystals and the problem of diffraction, Directions in Mathematical Quasicrustals, M. Baake and R. Moody, eds., CRM Monograph series, V.13, AMS, Providence RI, 2000, 61–93.

8. N. Lev, A. Olevskii, Quasicrystals and Poisson’s summation formula, Invent. Math., 200 (2015), 585– 606.

9. N. Lev, A. Olevskii, Fourier quasicrystals and discreteness of the difraction spectrum, arXive: 1512.08735v1[math.CA], 2015.

10. Y. Meyer, Quasicrystals, almost periodic patterns, mean–periodic functions, and irregular sampling, African Diaspora Journal of Mathematics, 13 (2012), №1, 1–45.

11. Y. Meyer, Nombres de Pizot, Nombres de Salem et analyse harmonique, Lect. Notes Math., 117 (1970), Springer-Verlag, 25 p.

12. R.V. Moody, Meyer’s sets and their duals. – 403–442 in R.V. Moody, Ed. The Mathematics of Long- Range Order, NATO ASISeries C., Springer-Verlag, New York, 1997.

13. W. Rudin, Fourier analysis on groups. – Interscience Publications, a Division of John Wiley and Sons, 1962, 285 p.

14. L.I. Ronkin, Almost periodic distributions and divisors in tube domains, Zap. Nauchn. Sem. POMI, 247 (1997), 210–236. (in Russian)

15. H.P. Rosenthal, Projections onto translation-invariant subspace of $L^P(G)$, Memoirs Amer. Math. Soc., 63 (1966).

16. V.S. Vladimirov, Equations of mathematical physics. – Marcel Dekker, Inc., New-York, 1971, 418 p.

	Some properties of measures with discrete support
Author	S. Yu. Favorov sfavorov@gmail.com Karazin’s Kharkiv National University
Abstract	We give some new conditions for the support of a discrete measure on Euclidean space to be a finite union of translated lattices. In particular, we consider the case when values of masses $a_{\lambda}$ of discrete measure satisfy the equality $G(a_{\lambda},\bar a_{\lambda})=0$ for each analytic function $G(z,w)$.
Keywords	distribution; Fourier transform; measure with discrete support; spectrum of measure; almost periodic measure; lattice
DOI	doi:10.15330/ms.46.2.189-195
Reference	1. A. Cordoba, Dirac combs, Lett. Math. Phis., 17 (1989), 191–196. 2. P.J. Cohen, On a conjecture of Littlewood and idempotent meagures, Am. J. Math., 82 (1960), 191–212. 3. S.Yu. Favorov, Bohr and Besicovitch almost periodic discrete sets and quasicrystals, Proc. Amer. Math. Soc., 140 (2012), 1761–1767. 4. S.Yu. Favorov, Fourier quasicrystals and Lagarias’ conjecture, Proc. Amer. Math. Soc., 144 (2016), 3527–3536. 5. M.N. Kolountzakis, On the structure of multiple translations tilings by polygonal regions, Preprint, 1999, 16 p. 6. M.N. Kolountzakis, J.C. Lagarias, Structure of tilings of the line by a function, Duke Math. Journal, 82 (1996), 653–678. 7. J.C. Lagarias, Mathematical quasicrystals and the problem of diffraction, Directions in Mathematical Quasicrustals, M. Baake and R. Moody, eds., CRM Monograph series, V.13, AMS, Providence RI, 2000, 61–93. 8. N. Lev, A. Olevskii, Quasicrystals and Poisson’s summation formula, Invent. Math., 200 (2015), 585– 606. 9. N. Lev, A. Olevskii, Fourier quasicrystals and discreteness of the difraction spectrum, arXive: 1512.08735v1[math.CA], 2015. 10. Y. Meyer, Quasicrystals, almost periodic patterns, mean–periodic functions, and irregular sampling, African Diaspora Journal of Mathematics, 13 (2012), №1, 1–45. 11. Y. Meyer, Nombres de Pizot, Nombres de Salem et analyse harmonique, Lect. Notes Math., 117 (1970), Springer-Verlag, 25 p. 12. R.V. Moody, Meyer’s sets and their duals. – 403–442 in R.V. Moody, Ed. The Mathematics of Long- Range Order, NATO ASISeries C., Springer-Verlag, New York, 1997. 13. W. Rudin, Fourier analysis on groups. – Interscience Publications, a Division of John Wiley and Sons, 1962, 285 p. 14. L.I. Ronkin, Almost periodic distributions and divisors in tube domains, Zap. Nauchn. Sem. POMI, 247 (1997), 210–236. (in Russian) 15. H.P. Rosenthal, Projections onto translation-invariant subspace of $L^P(G)$, Memoirs Amer. Math. Soc., 63 (1966). 16. V.S. Vladimirov, Equations of mathematical physics. – Marcel Dekker, Inc., New-York, 1971, 418 p.
Pages	189-195
Volume	46
Issue	2
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html