Morten Nielsen

Aalborg University, Ålborg, North Denmark, Denmark

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Publications (53)27.94 Total impact

  • Morten Nielsen, Hrvoje Šikić
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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
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    Morten Nielsen
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    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.
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    Morten Nielsen, Hrvoje Šikić
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    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
    Collectanea Mathematica 01/2012; 63(2):195-202. · 0.79 Impact Factor
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    Morten Nielsen, Kenneth N. Rasmussen
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    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 01/2012; 18(1):87-117. · 1.08 Impact Factor
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    Rémi Gribonval, Morten Nielsen
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    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.
    02/2011;
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    Morten Nielsen
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    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
    Journal of Geometric Analysis 01/2011; 22(1):12-22. · 0.86 Impact Factor
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    Kenneth N Rasmussen, Morten Nielsen
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    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
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    Morten Nielsen
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    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 01/2010; · 1.05 Impact Factor
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    Morten Nielsen
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    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
    Collectanea Mathematica 01/2010; 61(2):173-190. · 0.79 Impact Factor
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    Morten Nielsen
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    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. · 1.08 Impact Factor
  • Rémi Gribonval, Morten Nielsen
    Adv. Comput. Math. 01/2008; 28:23-41.
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    Morten Nielsen, Hrvoje Šikić
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    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 01/2008; · 0.76 Impact Factor
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    Lasse Borup, Rémi Gribonval, Morten Nielsen
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    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;
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    Lasse Borup, Morten Nielsen
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    ABSTRACT: A new construction of tight frames for L2(\Bbb Rd)L_2({\Bbb R}^d) with flexible time-frequency 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 so-called 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, α-modulation spaces, and anisotropic α-modulation spaces are considered. Finally, curvelet-type 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):39-70. · 1.08 Impact Factor
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    Morten Nielsen
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    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
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    Morten Nielsen, Hrvoje Šikić
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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;
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    Morten Nielsen
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    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 Strang-Fix conditions of arbitrary order.
    Advances in Computational Mathematics 01/2007; 27:195-209. · 1.47 Impact Factor
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    Lasse Borup, Morten Nielsen
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    ABSTRACT: A construction of Triebel-Lizorkin type spaces associated with flexible de-compositions 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 struc-ture of the decomposition of the frequency space, and for a very particular choice of decomposition function the spaces are reduced to classical (anisotropic) Triebel-Lizorkin spaces. An explicit atomic decomposition of the Triebel-Lizorkin type spaces is provided, and their interpolation properties are studied. As the main application, we consider Hörmander type classes of pseudo-differential operators adapted to the anisotropy and boundedness of such operators between corresponding Triebel-Lizorkin type spaces is proved.
    Journal of Function Spaces and Applications. 07/2006; 6.
  • Rémi Gribonval, Morten Nielsen
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    ABSTRACT: We consider two approaches for sparse decomposition of polyphonic music: a time-domain approach based on a shift-invariant model, and a frequency-domain approach based on phase-invariant power spectra. When trained on an example of a MIDI-controlled ...
    Signal Processing 01/2006; 86:415-416. · 2.24 Impact Factor
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    Rémi Gribonval, Morten Nielsen
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    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):51-71, 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 block-diagonal 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

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