[Show abstract][Hide abstract] ABSTRACT: In this paper we present a construction of frames generated by a single band-limited function for decomposition smoothness spaces on of modulation and Triebel-Lizorkin type. A perturbation argument is then used to construct compactly supported frame generators.
[Show abstract][Hide abstract] ABSTRACT: We apply the Garnett–Jones distance to the analysis of Schauder bases of translates. A special role is played by periodization functions pψpψ with lnpψ in the closure of L∞L∞ in BMO(T)BMO(T). In particular, for Schauder bases with such periodization functions we study the corresponding coefficient space. We also use the Garnett–Jones distance approach to show the stability of bases of translates with respect to convolution powers. The case of democratic conditional Schauder bases of translates is emphasized, as well.
[Show abstract][Hide abstract] ABSTRACT: The decomposition space approach is a general method to construct smoothness spaces on
$\mathbb{R }^d$
R
d
that include Besov, Triebel–Lizorkin, modulation, and
$\alpha $
α
-modulation spaces as special cases. This method also handles isotropic and an-isotropic spaces within the same framework. In this paper we consider a trace theorem for general decomposition type smoothness spaces. The result is based on a simple geometric estimate related to the structure of coverings of the frequency space used in the construction of decomposition spaces.
Monatshefte für Mathematik 09/2013; 171(3-4). DOI:10.1007/s00605-013-0532-z · 0.65 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted -spaces. We introduce a product Muckenhoupt condition for matrix weights and prove that rectangular Fourier partial sums converge in the corresponding matrix weighted space , , if and only if the weight satisfies the product Muckenhoupt condition. The same result is shown to hold true for other summation methods such as Cesàro and summation with the Jackson kernel.
[Show abstract][Hide abstract] ABSTRACT: We characterize Muckenhoupt A
p
weights in the product case on
\mathbbRN{\mathbb{R}^N} in terms of a graded family of A
p
conditions defined by rectangles with a lower bound on eccentricity. The connection to maximal functions and geometric coverings
is also studied.
KeywordsMaximal function–Product condition–Muckenhoupt weight
[Show abstract][Hide abstract] ABSTRACT: In this article we study a construction of compactly supported frame expansions for decomposition spaces of Triebel-Lizorkin
type and for the associated modulation spaces. This is done by showing that finite linear combinations of shifts and dilates
of a single function with sufficient decay in both direct and frequency space can constitute a frame for Triebel-Lizorkin
type spaces and the associated modulation spaces. First, we extend the machinery of almost diagonal matrices to Triebel-Lizorkin
type spaces and the associated modulation spaces. Next, we prove that two function systems which are sufficiently close have
an almost diagonal “change of frame coefficient” matrix. Finally, we approximate to an arbitrary degree an already known frame
for Triebel-Lizorkin type spaces and the associated modulation spaces with a single function with sufficient decay in both
direct and frequency space.
KeywordsDecomposition spaces–Anisotropic Triebel-Lizorkin spaces–Anisotropic Besov spaces–Frames
Journal of Fourier Analysis and Applications 02/2012; 18(1):87-117. DOI:10.1007/s00041-011-9190-5 · 1.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It is now well known that sparse or compressible vectors can be stably
recovered from their low-dimensional projection, provided the projection matrix
satisfies a Restricted Isometry Property (RIP). We establish new implications
of the RIP with respect to nonlinear approximation in a Hilbert space with a
redundant frame. The main ingredients of our approach are: a) Jackson and
Bernstein inequalities, associated to the characterization of certain
approximation spaces with interpolation spaces; b) a new proof that for
overcomplete frames which satisfy a Bernstein inequality, these interpolation
spaces are nothing but the collection of vectors admitting a representation in
the dictionary with compressible coefficients; c) the proof that the RIP
implies Bernstein inequalities. As a result, we obtain that in most
overcomplete random Gaussian dictionaries with fixed aspect ratio, just as in
any orthonormal basis, the error of best $m$-term approximation of a vector
decays at a certain rate if, and only if, the vector admits a compressible
expansion in the dictionary. Yet, for mildly overcomplete dictionaries with a
one-dimensional kernel, we give examples where the Bernstein inequality holds,
but the same inequality fails for even the smallest perturbation of the
dictionary.
Journal of Approximation Theory 02/2011; 165(1). DOI:10.1016/j.jat.2012.09.008 · 0.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider a periodic matrix weight W defined on ℝ
d
and taking values in the N×N positive-definite matrices. For such weights, we prove transference results between multiplier operators on L
p
(ℝ
d
;W) and
Lp(\mathbb Td;W)L_{p}(\mathbb {T}^{d};W), 1<p<∞, respectively. As a specific application, we study transference results for homogeneous multipliers of degree zero.
KeywordsTransference–Matrix weight–Muckenhoupt condition–Homogeneous multipliers
[Show abstract][Hide abstract] ABSTRACT: In this article we study a flexible method for constructing curvelet type frames. These curvelet type systems have the same sparse representation properties as curvelets for appropriate classes of smooth functions, and the flexibility of the method allows us to construct curvelet type systems with a prescribed nature such as compact support in direct space. The method consists of using the machinery of almost diagonal matrices to show that a system of curvelet molecules which is sufficiently close to curvelets constitutes a frame for curvelet type spaces. Such a system of curvelet molecules is then constructed using finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay.
Journal of Function Spaces and Applications 12/2010; 2012. DOI:10.1155/2012/876315 · 0.50 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We give a complete characterization of 2π-periodic matrix weights W for which the vector-valued trigonometric system forms a Schauder basis for the matrix weighted space Lp(T;W). Then trigonometric quasi-greedy bases for Lp(T;W) are considered. Quasi-greedy bases are systems for which the simple thresholding approximation algorithm converges in norm. It is proved that such a trigonometric basis can be quasi-greedy only for p=2, and whenever the system forms a quasi-greedy basis, the basis must actually be a Riesz basis.
Journal of Mathematical Analysis and Applications 11/2010; 371(2-371):784-792. DOI:10.1016/j.jmaa.2010.06.015 · 1.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We construct an orthonormal basis for the family of bi-variate α-modulation spaces. The construction is based on local trigonometric
bases, and the basis elements are closely related to so-called brushlets. As an application, we show thatm-term nonlinear approximation with the representing system in an α-modulation space can be completely characterized.
Keywordsα-modulation space-smoothness space-brushlets-local trigonometric bases-nonlinear approximation
MSC200041A17-42B35-42C15
[Show abstract][Hide abstract] ABSTRACT: We consider the problem of completely characterizing when a system of integer translates in a finitely generated shift-invariant
subspace of L
2(ℝ
d
) is stable in the sense that rectangular partial sums for the system are norm convergent. We prove that a system of integer
translates is stable in L
2(ℝ
d
) precisely when its associated Gram matrix satisfies a suitable Muckenhoupt A
2 condition.
KeywordsShift-invariant space-Schauder basis-Integer translates-Vector Hunt-Muckenhoupt-Wheeden theorem-Muckenhoupt condition
Mathematics Subject Classification (2000)41A45-42C15
Journal of Fourier Analysis and Applications 01/2010; 16(6):901-920. DOI:10.1007/s00041-009-9096-7 · 1.12 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We give a complete characterization of $2\pi$-periodic weights $w$ for which
the usual trigonometric system forms a quasi-greedy basis for $L^p(\bT;w)$,
i.e., bases for which simple thresholding approximants converge in norm. The
characterization implies that this can happen only for $p=2$ and whenever the
system forms a quasi-greedy basis, the basis must actually be a Riesz basis.
[Show abstract][Hide abstract] ABSTRACT: We consider a periodic matrix weight W defined on R,;W) converges if and only if W satisfies a matrix Muckenhoupt,Ap-condition.
[Show abstract][Hide abstract] ABSTRACT: We consider quasi-greedy systems of integer translates in a finitely generated shift-invariant subspace of L2(Rd), that is systems for which the thresholding approximation procedure is well behaved. We prove that every quasi-greedy system of integer translates is also a Riesz basis for its closed linear span. The result shows that there are no conditional quasi-greedy bases of integer translates in a finitely generated shift-invariant space.
Journal of Approximation Theory 11/2008; 155(1-155):43-51. DOI:10.1016/j.jat.2008.04.009 · 0.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We consider the problem of recovering a structured sparse representation of a signal in an overcomplete time–frequency dictionary with a particular structure. For infinite dictionaries that are the union of a nice wavelet basis and a Wilson basis, sufficient conditions are given for the basis pursuit and (orthogonal) matching pursuit algorithms to recover a structured representation of an admissible signal. The sufficient conditions take into account the structure of the wavelet/Wilson dictionary and allow very large (even infinite) support sets to be recovered even though the dictionary is highly coherent.
[Show abstract][Hide abstract] ABSTRACT: Finding a sparse approximation of a signal from an arbitrary dictionary is a very useful tool to solve many problems in signal processing. Several algorithms, such as Basis Pursuit (BP) and Matching Pursuits (MP, also known as greedy algorithms), have been introduced to compute sparse approximations of signals, but such algorithms a priori only provide sub-optimal solutions. In general, it is difficult to estimate how close a computed solution is from the optimal one. In a series of recent results, several authors have shown that both BP and MP can successfully recover a sparse representation of a signal provided that it is sparse enough, that is to say if its support (which indicates where are located the nonzero coefficients) is of sufficiently small size. In this paper we define identifiable structures that support signals that can be recovered exactly by L1 minimization (Basis Pursuit) and greedy algorithms. In other words, if the support of a representation belongs to an identifiable structure, then the representation will be recovered by BP and MP. In addition, we obtain that if the output of an arbitrary decomposition algorithm is supported on an identifiable structure, then one can be sure that the representation is optimal within the class of signals supported by the structure. As an application of the theoretical results, we give a detailed study of a family of multichannel dictionaries with a special structure (corresponding to the representation problem X = AS Phi^T ) often used in, e.g., under-determined source separation problems or in multichannel signal processing. An identifiable structure for such dictionaries is defined using a generalization of Tropp's Babel function which combines the coherence of the mixing matrix A with that of the time-domain dictionary Phi, and we obtain explicit structure conditions which ensure that both L1 minimization and a multichannel variant of Matching Pursuit can recover structured multichannel representations. The multichannel Matching Pursuit algorithm is described in detail and we conclude with a discussion of some implications of our results in terms of blind source separation based on sparse decompositions
[Show abstract][Hide abstract] ABSTRACT: Finding a sparse approximation of a signal from an arbitrary dictionary is a very useful tool to solve many problems in signal processing. Several algorithms, such as Basis Pursuit (BP) and Matching Pursuits (MP, also known as greedy algorithms), have been introduced to compute sparse approximations of signals, but such algorithms a priori only provide sub-optimal solutions. In general, it is difficult to estimate how close a computed solution is from the optimal one. In a series of recent results, several authors have shown that both BP and MP can successfully recover a sparse representation of a signal provided that it is sparse enough, that is to say if its support (which indicates where are located the nonzero coefficients) is of sufficiently small size. In this paper we define identifiable structures that support signals that can be recovered exactly by ℓ1 minimization (Basis Pursuit) and greedy algorithms. In other words, if the support of a representation belongs to an identifiable structure, then the representation will be recovered by BP and MP. In addition, we obtain that if the output of an arbitrary decomposition algorithm is supported on an identifiable structure, then one can be sure that the representation is optimal within the class of signals supported by the structure. As an application of the theoretical results, we give a detailed study of a family of multichannel dictionaries with a special structure (corresponding to the representation problem X = ASΦT) often used in, e.g., under-determined source sepa-ration problems or in multichannel signal processing. An identifiable structure for such dictionaries is defined using a generalization of Tropp's Babel function which combines the coherence of the mixing matrix A with that of the time-domain dictionary Φ, and we obtain explicit structure conditions which ensure that both ℓ1 minimization and a multi-channel variant of Matching Pursuit can recover structured multichannel representations. The multichannel Matching Pursuit algorithm is described in detail and we conclude with a discussion of some implications of our results in terms of blind source separation based on sparse decompositions.
[Show abstract][Hide abstract] ABSTRACT: We construct a uniformly bounded orthonormal almost greedy basis for Lp(0,1), 1<p<∞. The example shows that it is not possible to extend Orlicz's theorem, stating that there are no uniformly bounded orthonormal unconditional bases for Lp(0,1), p≠2, to the class of almost greedy bases.
Journal of Approximation Theory 12/2007; 149(2-149):188-192. DOI:10.1016/j.jat.2007.04.011 · 0.95 Impact Factor