Nonlinear Approximation with Dictionaries. II. Inverse Estimates

In this paper, which is the sequel to [16], we study inverse estimates of the Bernstein type for nonlinear approximation with structured redundant dictionaries in a Banach space. The main results are for blockwise incoherent dictionaries in Hilbert spaces, which generalize the notion of joint block-diagonal mutually incoherent bases introduced by Donoho and Huo. The Bernstein inequality obtained for such dictionaries is proved to be sharp, but it has an exponent that does not match that of the corresponding Jackson inequality.

[1]  E. M. Hartwell.  Boston , 1906 .

[2]  S. B. Stechkin.  On absolute convergence of orthogonal series. I. , 1953 .

[3]  H. Triebel.  Theory Of Function Spaces , 1983 .

[4]  C. Bennett,et al.  Interpolation of operators , 1987 .

[5]  P. Petrushev.  Direct and converse theorems for spline and rational approximation and besov spaces , 1988 .

[6]  B. Golubov,et al.  Walsh Series and Transforms , 1991 .

[7]  A. Efimov,et al.  Walsh Series and Transforms: Theory and Applications , 1991 .

[8]  Mladen Victor Wickerhauser.  Smooth localized orthonormal bases , 1993 .

[9]  G. Weiss,et al.  Local sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets , 1993 .

[10]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[11]  Ronald A. DeVore,et al.  Some remarks on greedy algorithms , 1996, Adv. Comput. Math..

[12]  Martin Greiner,et al.  Wavelets , 2018, Complex..

[13]  A. Calderbank,et al.  Z4‐Kerdock Codes, Orthogonal Spreads, and Extremal Euclidean Line‐Sets , 1997 .

[14]  P. Wojtaszczyk.  On unconditional polynomial bases in L p and bergman spaces , 1997 .

[15]  V. Temlyakov,et al.  Greedy Algorithms with Regard to Multivariate Systems with Special Structure , 1997 .

[16]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[17]  K. Gröchenig,et al.  Nonlinear Approximation with Local Fourier Bases , 2000 .

[18]  P. Tseng,et al.  Block Coordinate Relaxation Methods for Nonparametric Wavelet Denoising , 2000 .

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

[20]  A. Aldroubi,et al.  p-Frames and Shift Invariant Subspaces of Lp , 2001 .

[21]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[22]  R. Gribonval,et al.  Some remarks on non-linear approximation with Schauder bases , 2001 .

[23]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[24]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[25]  Stephen J. Dilworth,et al.  The Thresholding Greedy Algorithm, Greedy Bases, and Duality , 2003 .

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

[27]  Gerard Kerkyacharian,et al.  Entropy, Universal Coding, Approximation, and Bases Properties , 2003 .

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

[29]  Rémi Gribonval,et al.  Sparse representations in unions of bases , 2003, IEEE Trans. Inf. Theory.

[30]  Karlheinz Gröchenig,et al.  Localized Frames Are Finite Unions of Riesz Sequences , 2003, Adv. Comput. Math..

[31]  Karlheinz Gr öchenig.  Localization of Frames , 2004 .

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

[33]  R. Gribonval,et al.  On Approximation with Spline Generated Framelets , 2004 .

[34]  R. Gribonval,et al.  On a Problem of Gröchenig About Nonlinear Approximation with Localized Frames , 2004 .

[35]  R. Gribonval,et al.  Nonlinear Approximation with Dictionaries I. Direct Estimates , 2004 .