Vanishing Moment

14,000,000 Leading Edge Experts on the ideXlab platform

Scan Science and Technology

Contact Leading Edge Experts & Companies

Scan Science and Technology

Contact Leading Edge Experts & Companies

The Experts below are selected from a list of 3810 Experts worldwide ranked by ideXlab platform

Hartmut Führ - One of the best experts on this subject based on the ideXlab platform.

  • Simplified Vanishing Moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups
    Applied and Computational Harmonic Analysis, 2017
    Co-Authors: Hartmut Führ, Reihaneh Raisi Tousi
    Abstract:

    Abstract We consider the coorbit theory associated to a square-integrable, irreducible quasi-regular representation of a semidirect product group G = R d ⋊ H . The existence of coorbit spaces for this very general setting has been recently established, together with concrete Vanishing Moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups in arbitrary dimensions, as well as a class called generalized shearlet dilation groups, containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups are found. As a result, the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved.

  • Vanishing Moment conditions for wavelet atoms in higher dimensions
    Advances in Computational Mathematics, 2015
    Co-Authors: Hartmut Führ
    Abstract:

    We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in L2(?d), but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of GL(?d), the so-called dilation group.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Grochenig, and they can be informally described as follows: Given a function ? ? L2(?d) satisfying fairly mild decay, smoothness and Vanishing Moment conditions, any sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation.

  • Simplified Vanishing Moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups
    arXiv: Functional Analysis, 2014
    Co-Authors: Hartmut Führ, Reihaneh Raisi Tousi
    Abstract:

    We consider the coorbit theory associated to general continuous wavelet transforms arising from a square-integrable, irreducible quasi-regular representation of a semidirect product group $G = \mathbb{R}^d \rtimes H$. The existence of coorbit spaces for this very general setting has been recently established, together with concrete Vanishing Moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups, as well as a class called {\em generalized shearlet dilation groups}, containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups arise. As a result the existence of Banach frames consisting of compactly supported wavelets, with simultaneous convergence in a whole range of coorbit spaces, is established for all groups involved.

  • Vanishing Moment conditions for wavelet atoms in higher dimensions
    arXiv: Functional Analysis, 2013
    Co-Authors: Hartmut Führ
    Abstract:

    We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in ${\rm L}^2(\mathbb{R}^d)$, but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of ${\rm GL}(\mathbb{R}^d)$, the so-called {\em dilation group}.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Gr\"ochenig, and they can be informally described as follows: Given a function $\psi \in {\rm L}^2(\mathbb{R}^d)$ satisfying fairly mild decay, smoothness and Vanishing Moment conditions, {\em any} sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation.

Michael D. Adams - One of the best experts on this subject based on the ideXlab platform.

Václav Finěk - One of the best experts on this subject based on the ideXlab platform.

  • Quadratic spline wavelets with short support satisfying homogeneous boundary conditions
    ETNA - Electronic Transactions on Numerical Analysis, 2018
    Co-Authors: Dana Černá, Václav Finěk
    Abstract:

    In this paper, we construct a new quadratic spline-wavelet basis on the interval and on the unit square satisfying homogeneous Dirichlet boundary conditions of the first order. The wavelets have one Vanishing Moment and the shortest support among quadratic spline wavelets with at least one Vanishing Moment adapted to the same type of boundary conditions. The stiffness matrices arising from the discretization of the second-order elliptic problems using the constructed wavelet basis have uniformly bounded condition numbers, and the condition numbers are small. We present some quantitative properties of the constructed basis. We provide numerical examples to show that the Galerkin method and the adaptive wavelet method using our wavelet basis require fewer iterations than methods with other quadratic spline wavelet bases. Moreover, due to the small support of the wavelets, when using these methods with the new wavelet basis, the system matrix is sparser, and thus one iteration requires a smaller number of floating point operations than for other quadratic spline wavelet bases.

  • Quadratic Spline Wavelets with Short Support for Fourth-Order Problems
    Results in Mathematics, 2014
    Co-Authors: Dana Černá, Václav Finěk
    Abstract:

    In the paper, we propose constructions of new quadratic spline-wavelet bases on the interval and the unit square satisfying homogeneous Dirichlet boundary conditions of the second order. The basis functions have small supports and wavelets have one Vanishing Moment. We show that stiffness matrices arising from discretization of the biharmonic problem using a constructed wavelet basis have uniformly bounded condition numbers and these condition numbers are very small.

Zheng-xing Cheng - One of the best experts on this subject based on the ideXlab platform.

I.w. Selesnick - One of the best experts on this subject based on the ideXlab platform.

  • Cardinal multiwavelets and the sampling theorem
    1999 IEEE International Conference on Acoustics Speech and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258), 1999
    Co-Authors: I.w. Selesnick
    Abstract:

    This paper considers the classical Shannon sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang (1993), for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal. They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which has only 1 Vanishing Moment and is not continuous. This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of orthogonal multiscaling functions that are simultaneously cardinal, of compact support, and have more than one Vanishing Moment. The scaling functions thereby support a Shannon-like sampling theorem. Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator-the projection of a function onto the scaling space is given by its samples.

  • ICASSP - Cardinal multiwavelets and the sampling theorem
    1999 IEEE International Conference on Acoustics Speech and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258), 1999
    Co-Authors: I.w. Selesnick
    Abstract:

    This paper considers the classical Shannon sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang (1993), for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal. They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which has only 1 Vanishing Moment and is not continuous. This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of orthogonal multiscaling functions that are simultaneously cardinal, of compact support, and have more than one Vanishing Moment. The scaling functions thereby support a Shannon-like sampling theorem. Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator-the projection of a function onto the scaling space is given by its samples.

Related terms

Vanishing Moment - 15 days free trial to Access Experts & Abstracts