Quasicrystals and Control Theory
Cross-fertilization between control theory and irregular sampling is illustrated by two examples. Salah Baouendi applauded new ideas and welcomed new cultures. Interdisciplinary research is crossing frontiers, as Baouendi did all along his life. Two examples of cross-fertilization between harmonic analysis and control theory will be discussed in this homage. In 1983 Jacques-Louis Lions raised a problem in control theory. The solution I gave was grounded on a theorem on trigonometric sums proved by Arne Beurling. This will be our first example. The second example goes the other way around. A problem on trigonometric sums is solved using tools from control theory. Frontiers are erased as Baouendi wished.
KeywordsControl theory Quasicrystals Irregular sampling
2010 Mathematics Subject ClassificationPrimary 42A75 Secondary 94A20
I am very grateful to Linda Rothschild for her help and to the anonymous referee for her/his constructive criticism.
- 1.S.A. Avdonin, On the question of Riesz bases of exponential functions in \(L^2,\) Vestnik Leningrad. Univ. 13, 5-12 (1974). (Russian) English translation in Vestnik Leningrad Univ. Math. 7, 203-211 (1979)Google Scholar
- 3.A. Beurling, Balayage of Fourier-Stieltjes Transforms, in the Collected Works of Arne Beurling Harmonic Analysis, vol. 2 (Birkhäuser, Boston, 1989)Google Scholar
- 4.S. Grepstad, N. Lev, Universal sampling, quasicrystals and bounded remainder sets. C. R. Math. Acad. Sci. Paris (to appear)Google Scholar
- 5.S. Grepstad, N. Lev, Sets of bounded discrepancy for multi-dimensional irrational rotation, preprint (2014), arXiv:1404.0165
- 9.J-L. Lions, Sur le contrôle ponctuel de systèmes hyperboliques ou de type Petrowski, in Séminaire Equations aux dérivées partielles (Polytechnique), exp. no. 20, pp. 1–20 (1983–1984)Google Scholar
- 16.Y. Meyer, Addendum to Quasicrystals, almost periodic patterns, mean periodic functions and irregular sampling, Afr. Diaspora J. Math., to appearGoogle Scholar
- 18.Y. Meyer, Etude d’un modèle mathématique issu du contrôle des structures spatiales déformables. Nonlinear partial differential equations and their applications, in Collège de France Seminar, vol. VII, ed by H. Brezis, J.L. Lions Pitnam pp. 234–242 (1985)Google Scholar