Affine Density in Wavelet Analysis

Wavelet and Gabor Frames.- Weighted Affine Density.- Qualitative Density Conditions.- Quantitative Density Conditions.- Homogeneous Approximation Property.- Weighted Beurling Density and Shift-Invariant Gabor Systems.

[1]  Yang Wang Sparse complete Gabor systems on a lattice , 2004 .

[2]  J. Ramanathan,et al.  Incompleteness of Sparse Coherent States , 1995 .

[3]  C. Chui,et al.  Characterization of General Tight Wavelet Frames with Matrix Dilations and Tightness Preserving Oversampling , 2002 .

[4]  A. Olevskiǐ,et al.  Almost Integer Translates. Do Nice Generators Exist? , 2004 .

[5]  A. Ron,et al.  Weyl-Heisenberg frames and Riesz bases in $L_2(\mathbb{R}^d)$ , 1997 .

[6]  P. Casazza THE ART OF FRAME THEORY , 1999, math/9910168.

[7]  R. Balan,et al.  Density, overcompleteness, and localization of frames , 2006 .

[8]  D. Walnut,et al.  Differentiation and the Balian-Low Theorem , 1994 .

[9]  Wenchang Sun,et al.  Irregular Gabor frames and their stability , 2002 .

[10]  Demetrio Labate,et al.  A unified characterization of reproducing systems generated by a finite family, II , 2002 .

[11]  C. Heil,et al.  Density of frames and Schauder bases of windowed exponentials , 2008 .

[12]  Vadim Malyshev,et al.  Asymptotic combinatorics with application to mathematical physics , 2002 .

[13]  H. Feichtinger,et al.  Banach spaces related to integrable group representations and their atomic decompositions. Part II , 1989 .

[14]  Peter G. Casazza,et al.  Gabor Frames over Irregular Lattices , 2003, Adv. Comput. Math..

[15]  Walter Schachermayer,et al.  Stochastic Methods in Finance , 2004 .

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

[17]  H. Feichtinger,et al.  Banach Spaces of Distributions Defined by Decomposition Methods, I , 1985 .

[18]  Wojciech Czaja,et al.  The geometry of sets of parameters of wave packet frames , 2006 .

[19]  H. Feichtinger Generalized Amalgams, With Applications to Fourier Transform , 1990, Canadian Journal of Mathematics.

[20]  Gitta Kutyniok,et al.  The local integrability condition for wavelet frames , 2006 .

[21]  G. Weiss,et al.  A First Course on Wavelets , 1996 .

[22]  T. Strohmer,et al.  Efficient numerical methods in non-uniform sampling theory , 1995 .

[23]  Wenchang Sun,et al.  Density and stability of wavelet frames , 2003 .

[24]  Thomas Strohmer,et al.  GRASSMANNIAN FRAMES WITH APPLICATIONS TO CODING AND COMMUNICATION , 2003, math/0301135.

[25]  I. Daubechies,et al.  Gabor Time-Frequency Lattices and the Wexler-Raz Identity , 1994 .

[26]  R. Balan Stability theorems for Fourier frames and wavelet Riesz bases , 1997 .

[27]  Charles K. Chui,et al.  An Introduction to Wavelets , 1992 .

[28]  L. Baggett,et al.  Processing a radar signal and representations of the discrete Heisenberg group , 1990 .

[29]  K. Gröchenig Irregular sampling of wavelet and short-time Fourier transforms , 1993 .

[30]  Dennis Gabor,et al.  Theory of communication , 1946 .

[31]  O. Christensen Frames, Riesz bases, and discrete Gabor/wavelet expansions , 2001 .

[32]  H. Feichtinger On a new Segal algebra , 1981 .

[33]  C. Chui,et al.  Inequalities of Littlewood-Paley type for frames and wavelets , 1993 .

[34]  H. Feichtinger,et al.  Varying the time-frequency lattice of Gabor frames , 2003 .

[35]  S. Mallat A wavelet tour of signal processing , 1998 .

[36]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[37]  Wenchang Sun,et al.  Density of irregular wavelet frames , 2004 .

[38]  C. Chui Wavelets: A Tutorial in Theory and Applications , 1992 .

[39]  Karlheinz Gröchenig,et al.  Foundations of Time-Frequency Analysis , 2000, Applied and numerical harmonic analysis.

[40]  Kenneth R. Meyer,et al.  Periodic Solutions of the N-Body Problem , 2000 .

[41]  David R. Larson,et al.  Wavelet sets in ℝn , 1997 .

[42]  C. Heil History and Evolution of the Density Theorem for Gabor Frames , 2007 .

[43]  Y. Meyer Wavelets and Operators , 1993 .

[44]  C. Chui,et al.  Orthonormal wavelets and tight frames with arbitrary real dilations , 2000 .

[45]  S. Cerrai Second Order Pde's in Finite and Infinite Dimension: A Probabilistic Approach , 2001 .

[46]  Charles K. Chui,et al.  Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments , 2003, Adv. Comput. Math..

[47]  8 - Affine, Quasi-Affine and Co-Affine Wavelets , 2003 .

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

[49]  A. Olevskiǐ Completeness in L2(ℝ) of almost integer translates , 1997 .

[50]  A. Ron,et al.  Tight compactly supported wavelet frames of arbitrarily high smoothness , 1998 .

[51]  A. Grossmann,et al.  TRANSFORMS ASSOCIATED TO SQUARE INTEGRABLE GROUP REPRESENTATION. 2. EXAMPLES , 1986 .

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

[53]  A. Ron,et al.  Generalized Shift-Invariant Systems , 2005 .

[54]  I. Daubechies,et al.  Framelets: MRA-based constructions of wavelet frames☆☆☆ , 2003 .

[55]  C. Heil An Introduction to Weighted Wiener Amalgams , 2003 .

[56]  Gitta Kutyniok,et al.  Affine Density, Frame Bounds, and the Admissibility Condition for Wavelet Frames , 2007 .

[57]  H. Triebel Theory of Function Spaces III , 2008 .

[58]  J. Lagarias,et al.  Structure of tilings of the line by a function , 1996 .

[59]  Brody Dylan Johnson,et al.  On the Oversampling of Affine Wavelet Frames , 2003, SIAM J. Math. Anal..

[60]  K. Gröchenig Localization of Frames, Banach Frames, and the Invertibility of the Frame Operator , 2004 .

[61]  D. Speegle On the existence of wavelets for non-expansive dilation matrices , 2003 .

[62]  Charles K. Chui,et al.  Affine frames, quasi-affine frames, and their duals , 1998, Adv. Comput. Math..

[63]  Gitta Kutyniok,et al.  The Homogeneous Approximation Property for wavelet frames , 2007, J. Approx. Theory.

[64]  Peder A. Olsen,et al.  A note on irregular discrete wavelet transforms , 1992, IEEE Trans. Inf. Theory.

[65]  Wenchang Sun Density of wavelet frames , 2007 .

[66]  Nizar Touzi,et al.  Paris-Princeton Lectures on Mathematical Finance 2002 , 2003 .

[67]  G. Kutyniok,et al.  Density of weighted wavelet frames , 2003 .

[68]  Vivek K Goyal,et al.  Quantized Frame Expansions with Erasures , 2001 .

[69]  H. Landau On the density of phase-space expansions , 1993, IEEE Trans. Inf. Theory.

[70]  Eric Lombardi,et al.  Oscillatory Integrals and Phenomena Beyond all Algebraic Orders: with Applications to Homoclinic Orbits in Reversible Systems , 2000 .

[71]  L. Hörmander,et al.  An introduction to complex analysis in several variables , 1973 .

[72]  E. C. Titchmarsh On Conjugate Functions , 1929 .

[73]  Peter G. Casazza,et al.  Weyl-Heisenberg frames, translation invariant systems and the Walnut representation , 1999, math/9910169.

[74]  G. Folland A course in abstract harmonic analysis , 1995 .

[75]  Wenchang Sun,et al.  Irregular wavelet/Gabor frames , 2002 .

[76]  Ajem Guido Janssen,et al.  Signal Analytic Proofs of Two Basic Results on Lattice Expansions , 1994 .

[77]  I. Singer,et al.  Bases in Banach Spaces II , 1970 .

[78]  Pierre Bernard,et al.  Lectures on Probability Theory and Statistics: Ecole d'Ete de Probabilites de Saint-Flour XXVI - 1996 , 1997 .

[79]  H. Feichtinger,et al.  Banach spaces related to integrable group representations and their atomic decompositions, I , 1989 .

[80]  Olivier Catoni,et al.  Statistical learning theory and stochastic optimization , 2004 .

[81]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[82]  Demetrio Labate,et al.  Some remarks on the unified characterization of reproducing systems , 2005 .

[83]  C. Chui,et al.  Compactly supported tight and sibling frames with maximum vanishing moments , 2001 .

[84]  Stéphane Jaffard,et al.  A density criterion for frames of complex exponentials. , 1991 .

[85]  G. Kutyniok,et al.  Beurling Dimension of Gabor Pseudoframes for Affine Subspaces , 2008 .

[86]  E. Hewitt,et al.  Abstract Harmonic Analysis , 1963 .

[87]  Joram Lindenstrauss,et al.  Classical Banach spaces , 1973 .

[88]  K. Seip Beurling type density theorems in the unit disk , 1993 .

[89]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[90]  Karlheinz Gröchenig,et al.  On Landau's Necessary Density Conditions for Sampling and Interpolation of Band-Limited Functions , 1996 .

[91]  O. Christensen,et al.  Irregular Wavelet Frames and Gabor Frames , 2001 .

[92]  O. Christensen,et al.  Density of Gabor Frames , 1999 .

[93]  R. Balan,et al.  Density, Overcompleteness, and Localization of Frames. II. Gabor Systems , 2005 .

[94]  R. Chan,et al.  Tight frame: an efficient way for high-resolution image reconstruction , 2004 .

[95]  M. Rieffel Von Neumann algebras associated with pairs of lattices in Lie groups , 1981 .

[96]  K. Seip Density theorems for sampling and interpolation in the Bargmann-Fock space I , 1992, math/9204238.

[97]  Demetrio Labate,et al.  Oversampling, quasi-affine frames, and wave packets , 2004 .

[98]  Marcin Bownik,et al.  The Spectral Function of Shift-Invariant Spaces , 2003 .

[99]  D. Larson,et al.  Wandering Vectors for Unitary Systems and Orthogonal Wavelets , 1998 .

[100]  Olivier Pironneau,et al.  Optimal Shape Design , 2000 .

[101]  B. D. Johnson On the relationship between quasi-affine systems and the ` a trous algorithm , 2002 .

[102]  Christopher Heil,et al.  Continuous and Discrete Wavelet Transforms , 1989, SIAM Rev..

[103]  John J. Benedetto,et al.  Sigma-delta quantization and finite frames , 2004, ICASSP.