Publications (53)27.94 Total impact
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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.Journal of Functional Analysis 01/2014; 266(4):2281–2293. · 1.25 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.Journal of Mathematics. 03/2013; 2013.  [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 weightCollectanea Mathematica 01/2012; 63(2):195202. · 0.79 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this article we study a construction of compactly supported frame expansions for decomposition spaces of TriebelLizorkin 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 TriebelLizorkin type spaces and the associated modulation spaces. First, we extend the machinery of almost diagonal matrices to TriebelLizorkin 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 TriebelLizorkin type spaces and the associated modulation spaces with a single function with sufficient decay in both direct and frequency space. KeywordsDecomposition spaces–Anisotropic TriebelLizorkin spaces–Anisotropic Besov spaces–FramesJournal of Fourier Analysis and Applications 01/2012; 18(1):87117. · 1.08 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: It is now well known that sparse or compressible vectors can be stably recovered from their lowdimensional 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 onedimensional kernel, we give examples where the Bernstein inequality holds, but the same inequality fails for even the smallest perturbation of the dictionary.02/2011;  [Show abstract] [Hide abstract]
ABSTRACT: We consider a periodic matrix weight W defined on ℝ d and taking values in the N×N positivedefinite 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 multipliersJournal of Geometric Analysis 01/2011; 22(1):1222. · 0.86 Impact Factor  [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. · 0.50 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We give a complete characterization of 2πperiodic matrix weights W for which the vectorvalued trigonometric system forms a Schauder basis for the matrix weighted space Lp(T;W). Then trigonometric quasigreedy bases for Lp(T;W) are considered. Quasigreedy bases are systems for which the simple thresholding approximation algorithm converges in norm. It is proved that such a trigonometric basis can be quasigreedy only for p=2, and whenever the system forms a quasigreedy basis, the basis must actually be a Riesz basis.Journal of Mathematical Analysis and Applications 01/2010; · 1.05 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We construct an orthonormal basis for the family of bivariate αmodulation spaces. The construction is based on local trigonometric bases, and the basis elements are closely related to socalled brushlets. As an application, we show thatmterm nonlinear approximation with the representing system in an αmodulation space can be completely characterized. Keywordsαmodulation spacesmoothness spacebrushletslocal trigonometric basesnonlinear approximation MSC200041A1742B3542C15Collectanea Mathematica 01/2010; 61(2):173190. · 0.79 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We consider the problem of completely characterizing when a system of integer translates in a finitely generated shiftinvariant 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. KeywordsShiftinvariant spaceSchauder basisInteger translatesVector HuntMuckenhouptWheeden theoremMuckenhoupt condition Mathematics Subject Classification (2000)41A4542C15Journal of Fourier Analysis and Applications 01/2010; 16(6):901920. · 1.08 Impact Factor  Adv. Comput. Math. 01/2008; 28:2341.
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ABSTRACT: We consider quasigreedy systems of integer translates in a finitely generated shiftinvariant subspace of L2(Rd), that is systems for which the thresholding approximation procedure is well behaved. We prove that every quasigreedy system of integer translates is also a Riesz basis for its closed linear span. The result shows that there are no conditional quasigreedy bases of integer translates in a finitely generated shiftinvariant space.Journal of Approximation Theory 01/2008; · 0.76 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.Applied and Computational Harmonic Analysis. 01/2008;  [Show abstract] [Hide abstract]
ABSTRACT: A new construction of tight frames for L2(\Bbb Rd)L_2({\Bbb R}^d) with flexible timefrequency localization is considered. The frames can be adapted to form atomic decompositions for a large family of smoothness spaces on \Bbb Rd,{\Bbb R}^d, a class of socalled decomposition spaces. The decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients. As examples of the general construction, new tight frames yielding decompositions of Besov space, anisotropic Besov spaces, αmodulation spaces, and anisotropic αmodulation spaces are considered. Finally, curvelettype tight frames are constructed on \Bbb Rd, d ³ 2.{\Bbb R}^d, d \geq 2.Journal of Fourier Analysis and Applications 01/2007; 13(1):3970. · 1.08 Impact Factor  [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 01/2007; · 0.76 Impact Factor 
Article: Schauder bases of integer translates
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ABSTRACT: For a function Ψ∈L2(R), we give necessary and sufficient conditions for the family to be a Schauder basis for the space .Applied and Computational Harmonic Analysis. 01/2007;  [Show abstract] [Hide abstract]
ABSTRACT: Given a dilation matrix A: Z,), we consider polynomial solutions M to the equation P g2G M( + g) = 1 with the constraints that M,0 and M(0) = 1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying StrangFix conditions of arbitrary order.Advances in Computational Mathematics 01/2007; 27:195209. · 1.47 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: A construction of TriebelLizorkin type spaces associated with flexible decompositions of the frequency space R d is considered. The class of admissible frequency decompositions is generated by a one parameter group of (anisotropic) dilations on R d and a suitable decomposition function. The decomposition function governs the structure of the decomposition of the frequency space, and for a very particular choice of decomposition function the spaces are reduced to classical (anisotropic) TriebelLizorkin spaces. An explicit atomic decomposition of the TriebelLizorkin type spaces is provided, and their interpolation properties are studied. As the main application, we consider Hörmander type classes of pseudodifferential operators adapted to the anisotropy and boundedness of such operators between corresponding TriebelLizorkin type spaces is proved.Journal of Function Spaces and Applications. 07/2006; 6.  [Show abstract] [Hide abstract]
ABSTRACT: We consider two approaches for sparse decomposition of polyphonic music: a timedomain approach based on a shiftinvariant model, and a frequencydomain approach based on phaseinvariant power spectra. When trained on an example of a MIDIcontrolled ...Signal Processing 01/2006; 86:415416. · 2.24 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, which is the sequel to [R. Gribonval and M. Nielsen. Nonlinear approximation with dictionaries. I. Direct estimates. J. Fourier Anal. Appl., 10(1):5171, 2004], we study inverse esti mates of the Bernstein type for nonlinear approximation with structured redun dant dictionaries in a Banach space. The main results are for blockwise incoherent dictionaries in Hilbert spaces, which generalize the notion of joint blockdiagonal mutually incoherent bases introduced by Donoho and Huo. The Bernstein inequal ity obtained for such dictionaries is proved to be sharp, but it has an exponent that does not match that of the corresponding Jackson inequality.Constructive Approximation 01/2006; · 1.07 Impact Factor
Publication Stats
379  Citations  
27.94  Total Impact Points  
Top Journals
Institutions

2003–2014

Aalborg University
 Department of Mathematical Sciences
Ålborg, North Denmark, Denmark


2007–2008

University of Zagreb
 Department of Mathematics
Zagrabia, Grad Zagreb, Croatia


2006

Washington University in St. Louis
San Luis, Missouri, United States


2000–2002

University of South Carolina
 Department of Mathematics
Columbia, South Carolina, United States
