Model Sets and New Versions of Shannon Sampling Theorem

In these notes distinct approaches to define model sets/quasicrystals are discussed. We also discuss some improvements on Shannon sampling theorem obtained by using simple model sets/quasicrystals.

[1]  A. Olevskiǐ,et al.  Universal sampling of band-limited signals , 2006 .

[2]  M. Senechal Quasicrystals and geometry , 1995 .

[3]  Y. Meyer Trois problèmes sur les sommes trigonométriques , 1973 .

[4]  Robert V. Moody,et al.  Model Sets: A Survey , 2000 .

[5]  R. Salem Algebraic numbers and Fourier analysis , 1963 .

[6]  Yves Meyer Le spectre de Wiener , 1966 .

[7]  A. Olevskiǐ,et al.  Universal Sampling and Interpolation of Band-Limited Signals , 2008 .

[8]  Basarab Matei,et al.  A variant of compressed sensing , 2009 .

[9]  J. Cahn,et al.  Metallic Phase with Long-Range Orientational Order and No Translational Symmetry , 1984 .

[10]  Peter A. B. Pleasants,et al.  Repetitive Delone sets and quasicrystals , 2003, Ergodic Theory and Dynamical Systems.

[11]  Jeffrey C. Lagarias,et al.  Meyer's concept of quasicrystal and quasiregular sets , 1996 .

[12]  G. Kozma,et al.  Exponential Riesz Bases, Discrepancy of Irrational Rotations and BMO , 2010, 1009.2188.

[13]  Y. Meyer,et al.  Simple quasicrystals are sets of stable sampling , 2010 .

[14]  L. Schwartz Théorie des distributions , 1966 .

[15]  R. Moody Uniform Distribution in Model Sets , 2002, Canadian Mathematical Bulletin.

[16]  G. Kozma,et al.  Combining Riesz bases , 2012, 1210.6383.

[17]  At Hof,et al.  On diffraction by aperiodic structures , 1995 .

[18]  Jean-Pierre Kahane,et al.  Pseudo-périodicité et séries de Fourier lacunaires , 1962 .

[19]  R. Moody MATHEMATICAL QUASICRYSTALS: A TALE OF TWO TOPOLOGIES , 2006 .

[20]  Jeffrey C. Lagarias,et al.  Geometric Models for Quasicrystals I. Delone Sets of Finite Type , 1999, Discret. Comput. Geom..

[21]  Y. Meyer,et al.  Nombres de Pisot, Nombres de Salem et Analyse Harmonique , 1970 .

[22]  R. Moody Meyer Sets and Their Duals , 1997 .

[23]  A. Ingham Some trigonometrical inequalities with applications to the theory of series , 1936 .

[24]  Yves Meyer,et al.  Quasicrystals, Diophantine approximation and algebraic numbers , 1995 .

[25]  Karlheinz Gröchenig,et al.  Random Sampling of Entire Functions of Exponential Type in Several Variables , 2007 .

[26]  Jeffrey C. Lagarias,et al.  Geometric Models for Quasicrystals II. Local Rules Under Isometries , 1999, Discret. Comput. Geom..

[27]  H. Landau Necessary density conditions for sampling and interpolation of certain entire functions , 1967 .

[28]  N. Wiener The Fourier Integral: and certain of its Applications , 1933, Nature.

[29]  Basarab Matei,et al.  Quasicrystals are sets of stable sampling , 2008 .

[30]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[31]  Yves Meyer,et al.  Algebraic numbers and harmonic analysis , 1972 .

[32]  M. Baake,et al.  Mathematical quasicrystals and the problem of diffraction , 2000 .

[33]  G. Chistyakov,et al.  Random perturbations of exponential Riesz bases in $L^2(-\pi,\pi)$ , 1997 .