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Charles A. Micchelli - One of the best experts on this subject based on the ideXlab platform.

  • regularity of multiwavelets
    Advances in Computational Mathematics, 1997
    Co-Authors: Charles A. Micchelli, Tomas Sauer
    Abstract:

    The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the Refinement Equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets.

  • Using the Matrix Refinement Equation for the Construction of Wavelets on Invariant Sets
    Applied and Computational Harmonic Analysis, 1994
    Co-Authors: Charles A. Micchelli
    Abstract:

    Abstract We construct discontinuous wavelets on invariant sets in R n by using the matrix Refinement Equation and the basic operation of translation and scale. In the special case of the unit interval we show how to modify our construction to obtain continuous wavelets.

  • Using the matrix Refinement Equation for the construction of wavelets II: smooth wavelets on [0,1]
    Approximation and Computation: A Festschrift in Honor of Walter Gautschi, 1994
    Co-Authors: Charles A. Micchelli
    Abstract:

    This paper continues the work in [4] on constructing orthogonal bases on the interval [0,1] by using the matrix Refinement Equation and the two basic operations of translation and scale. We call the elements of these bases wavelets. Here we amplify on the applicability of our method and construct smooth wavelets with and without boundary conditions. That is, we describe a procedure to recursively generate orthonormal bases with any prescribed number of continuous derivatives. As a caveat to the reader we reiterate our remark above that this paper is a continuation of our work in [4] and therefore some familiarity with [4] is assumed.

  • Orthonormal Cardinal Functions
    Wavelets: Theory Algorithms and Applications, 1994
    Co-Authors: Tim N. T. Goodman, Charles A. Micchelli
    Abstract:

    Abstract In this paper we investigate various questions relating to the construction of functions with one or more of the following properties: satisfying a Refinement Equation, orthonormal integer translates, compact support, band-limited, symmetric, or cardinal interpolatory.

  • Using the Refinement Equation for evaluating integrals of wavelets
    SIAM Journal on Numerical Analysis, 1993
    Co-Authors: Wolfgang Dahmen, Charles A. Micchelli
    Abstract:

    The Wavelet Galerkin Method for solving partial differential Equations leads to the problem of computing integrals of products of derivatives of wavelets. This paper studies the problem from the point of view of stationary subdivision schemes. One of the main results is to identify these integrals as components of the unique solution of a certain eigenvector-moment problem associated with the coefficients of the Refinement Equation. Asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach.

Qingtang Jiang - One of the best experts on this subject based on the ideXlab platform.

  • original article quad triangle subdivision nonhomogeneous Refinement Equation and polynomial reproduction
    Mathematics and Computers in Simulation, 2012
    Co-Authors: Qingtang Jiang
    Abstract:

    The quad/triangular subdivision, whose control net and refined meshes consist of both quads and triangles, provides better visual quality of subdivision surfaces. While some theoretical results such as polynomial reproduction and smoothness analysis of quad/triangle schemes have been obtained in the literature, some issues such as the basis functions at quad/triangle vertices and design of interpolatory quad/triangle schemes need further study. In our study of quad/triangle schemes, we observe that a quad/triangle subdivision scheme can be derived from a nonhomogeneous Refinement Equation. Hence, the basis functions at quad/triangle vertices are shifts of the refinable function associated with a nonhomogeneous Refinement Equation. In this paper a quad/triangle subdivision surface is expressed analytically as the linear combination of these basis functions and the polynomial reproduction of matrix-valued quad/triangle schemes is studied. The result on polynomial reproduction achieved here is critical for the smoothness analysis and construction of matrix-valued quad/triangle schemes. Several new schemes are also constructed.

  • Original Article: Quad/triangle subdivision, nonhomogeneous Refinement Equation and polynomial reproduction
    Mathematics and Computers in Simulation, 2012
    Co-Authors: Qingtang Jiang
    Abstract:

    The quad/triangular subdivision, whose control net and refined meshes consist of both quads and triangles, provides better visual quality of subdivision surfaces. While some theoretical results such as polynomial reproduction and smoothness analysis of quad/triangle schemes have been obtained in the literature, some issues such as the basis functions at quad/triangle vertices and design of interpolatory quad/triangle schemes need further study. In our study of quad/triangle schemes, we observe that a quad/triangle subdivision scheme can be derived from a nonhomogeneous Refinement Equation. Hence, the basis functions at quad/triangle vertices are shifts of the refinable function associated with a nonhomogeneous Refinement Equation. In this paper a quad/triangle subdivision surface is expressed analytically as the linear combination of these basis functions and the polynomial reproduction of matrix-valued quad/triangle schemes is studied. The result on polynomial reproduction achieved here is critical for the smoothness analysis and construction of matrix-valued quad/triangle schemes. Several new schemes are also constructed.

  • On Existence and Weak Stability of Matrix Refinable Functions
    Constructive Approximation, 1999
    Co-Authors: Qingtang Jiang, Zuowei Shen
    Abstract:

    We consider the existence of distributional (or L2 ) solutions of the matrix Refinement Equation

Ding-xuan Zhou - One of the best experts on this subject based on the ideXlab platform.

  • L _ p solutions of Refinement Equations
    Journal of Fourier Analysis and Applications, 2001
    Co-Authors: Rong-qing Jia, Ka-sing Lau, Ding-xuan Zhou
    Abstract:

    In the recent characterizations of the L_p solution of the Refinement Equation in terms of the “p-norm joint spectral radius,” there are problems in choosing the initial function for iteration [3, 23], or in addition, requiring stability of the refinable function [13, 17]. In this article we overcome these difficulties and give a more complete characterization of this nature. The criterion is constructive and can be implemented. It can be used to describe the regularity of the solution without assuming stability. This has significant advantages over the previous work. The corresponding results for vector Refinement Equations are also discussed.

  • Local linear independence of refinable vectors of functions
    Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2000
    Co-Authors: T. N. T. Goodman, Ding-xuan Zhou
    Abstract:

    This paper is devoted to a study of local linear independence of refinable vectors of functions. A vector of functions φ = (φ1, . . . , φr) ∈ (C(IR))r is said to be refinable if it satisfies the vector Refinement Equation φ(x) = ∑ α∈Z s a(α)φ(2x− α), where a is a finitely supported sequence of r×r matrices called the Refinement mask. A complete characterization for the local linear independence of the shifts of φ1, . . . , φr is given strictly in terms of the mask. Several examples are provided to illustrate the general theory. This investigation is important for construction of wavelets on bounded domains and nonlinear approximation by wavelets. Local Linear Independence of Refinable Vectors of Functions

  • Convergence of Subdivision Schemes Associated with Nonnegative Masks
    SIAM Journal on Matrix Analysis and Applications, 2000
    Co-Authors: Rong-qing Jia, Ding-xuan Zhou
    Abstract:

    This paper is concerned with Refinement Equations of the type $$ f = \sum_{\alpha\in\bbbz^s} a(\alpha) f({M\kern .12em\cdot}-\alpha), $$ where f is the unknown function defined on the s-dimensional Euclidean space $\bbbr^s$, a is a finitely supported sequence on $\bbbz^s$, and M is an s x s dilation matrix with m := |det M|. The solution of a Refinement Equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach.

  • Inhomogeneous Refinement Equations
    The Journal of Fourier Analysis and Applications, 1998
    Co-Authors: Gilbert Strang, Ding-xuan Zhou
    Abstract:

    Equations with two time scales (Refinement Equations or dilation Equations) are central to wavelet theory. Several applications also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous Refinement Equation $$\phi (t) = \sum\limits_{k \in \mathbb{Z}} {a(k)\phi (2t - k) + F(t)}$$ where a (k) is a finite sequence and F is a compactly supported distribution on ℝ.

Rong-qing Jia - One of the best experts on this subject based on the ideXlab platform.

  • Convergence rates of cascade algorithms
    Proceedings of the American Mathematical Society, 2003
    Co-Authors: Rong-qing Jia
    Abstract:

    We consider solutions of a Refinement Equation of the form Φ = γeZ a Σ a(γ)Φ(M.-γ), where a is a finitely supported sequence called the Refinement mask. Associated with the mask a is a linear operator Q a defined on L p (R s ) by Q a ψ:= γ ∈ Z s a(γ)ψ(M.-γ). This paper is concerned with the convergence of the cascade algorithm associated with a, i.e., the convergence of the sequence (Q n aψ) n=1,2,... in the L p -norm. Our main result gives estimates for the convergence rate of the cascade algorithm. Let Φ be the normalized solution of the above Refinement Equation with the dilation matrix M being isotropic. Suppose Φ lies in the Lipschitz space Lip(μ,L p (R s )), where μ > 0 and 1 < p < ∞. Under appropriate conditions on ψ, the following estimate will be established: ∥Q n aψ-Φ∥ p ≤C(m -1/s ) μn ∀n ∈ N, where m:= |det M| and C is a constant. In particular, we confirm a conjecture of A. Ron on convergence of cascade algorithms.

  • L _ p solutions of Refinement Equations
    Journal of Fourier Analysis and Applications, 2001
    Co-Authors: Rong-qing Jia, Ka-sing Lau, Ding-xuan Zhou
    Abstract:

    In the recent characterizations of the L_p solution of the Refinement Equation in terms of the “p-norm joint spectral radius,” there are problems in choosing the initial function for iteration [3, 23], or in addition, requiring stability of the refinable function [13, 17]. In this article we overcome these difficulties and give a more complete characterization of this nature. The criterion is constructive and can be implemented. It can be used to describe the regularity of the solution without assuming stability. This has significant advantages over the previous work. The corresponding results for vector Refinement Equations are also discussed.

  • Convergence of Subdivision Schemes Associated with Nonnegative Masks
    SIAM Journal on Matrix Analysis and Applications, 2000
    Co-Authors: Rong-qing Jia, Ding-xuan Zhou
    Abstract:

    This paper is concerned with Refinement Equations of the type $$ f = \sum_{\alpha\in\bbbz^s} a(\alpha) f({M\kern .12em\cdot}-\alpha), $$ where f is the unknown function defined on the s-dimensional Euclidean space $\bbbr^s$, a is a finitely supported sequence on $\bbbz^s$, and M is an s x s dilation matrix with m := |det M|. The solution of a Refinement Equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach.

  • Spectral properties of the transition operator associated to a multivariate Refinement Equation
    Linear Algebra and its Applications, 1999
    Co-Authors: Rong-qing Jia, Shurong Zhang
    Abstract:

    Given a finitely supported sequence a on Z s and an s s dilation matrix M, the transition operator Ta is the linear operator defined by TavOaU :a P b2Z s aOMabUvObU, where a2 Z s and v lies in '0OZ s U, the linear space of all finitely supported sequences on Z s . In this paper we investigate the spectral properties of the transition operator Ta and apply these properties to the study of the approximation and smoothness properties of the normalized solution of the Refinement Equation /a P a2Z s aOaU/OMaU . "1999

  • Stability and linear independence associated with wavelet decompositions
    Proceedings of the American Mathematical Society, 1993
    Co-Authors: Rong-qing Jia, Jianzhong Wang
    Abstract:

    Wavelet decompositions are based on basis functions satisfying Refinement Equations. The stability, linear independence, and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask sequence in the Refinement Equation that the basis function satisfies

Ursula Molter - One of the best experts on this subject based on the ideXlab platform.

  • Approximation by crystal-refinable functions
    Geometriae Dedicata, 2020
    Co-Authors: Ursula Molter, María Del Carmen Moure, Alejandro Quintero
    Abstract:

    Let $$\varGamma $$ Γ be a crystal group in $$\mathbb {R}^d$$ R d . A function $$\varphi :\mathbb {R}^d\longrightarrow \mathbb {C}$$ φ : R d ⟶ C is said to be crystal-refinable (or $$\varGamma $$ Γ -refinable) if it is a linear combination of finitely many of the rescaled and translated functions $$\varphi (\gamma ^{-1}(ax))$$ φ ( γ - 1 ( a x ) ) , where the translations $$\gamma $$ γ are taken on a crystal group $$\varGamma $$ Γ , and a is an expansive dilation matrix such that $$a\varGamma a^{-1}\subset \varGamma .$$ a Γ a - 1 ⊂ Γ . A $$\varGamma $$ Γ -refinable function $$\varphi : \mathbb {R}^d \rightarrow \mathbb {C}$$ φ : R d → C satisfies a Refinement Equation $$\varphi (x)=\sum _{\gamma \in \varGamma }d_\gamma \varphi (\gamma ^{-1}(ax))$$ φ ( x ) = ∑ γ ∈ Γ d γ φ ( γ - 1 ( a x ) ) with $$d_\gamma \in \mathbb {C}$$ d γ ∈ C . Let $$\mathcal S(\varphi )$$ S ( φ ) be the linear span of $$\{\varphi (\gamma ^{-1}(x)): \gamma \in \varGamma \}$$ { φ ( γ - 1 ( x ) ) : γ ∈ Γ } and $$\mathcal {S}^h=\{f(x/h):f\in \mathcal {S(\varphi )}\}$$ S h = { f ( x / h ) : f ∈ S ( φ ) } . One important property of $$\mathcal S(\varphi )$$ S ( φ ) is, how well it approximates functions in $$L^2(\mathbb {R}^d)$$ L 2 ( R d ) . This property is very closely related to the crystal-accuracy of $$\mathcal S(\varphi )$$ S ( φ ) , which is the highest degree p such that all multivariate polynomials q ( x ) of $$\mathrm{degree}(q)

  • REFINABLE SHIFT INVARIANT SPACES IN ℝd
    International Journal of Wavelets Multiresolution and Information Processing, 2005
    Co-Authors: Carlos Cabrelli, Sigrid B. Heineken, Ursula Molter
    Abstract:

    Let φ : ℝd → ℂ be a compactly supported function which satisfies a Refinement Equation of the form where Γ ⊂ ℝd is a lattice, Λ is a finite subset of Γ, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Γ-translates of φ, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of L = [cAi-j]i,j∈Γ and a finite-dimensional subspace in the shift-invariant space generated by φ. We provide a basis of and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function φ has accuracy κ, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than κ. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A-1. Furthermore we show that the dimension of coincides with the local dimension of φ, and hence, every function in the shift-invariant space generated by φ can be written locally as a linear combination of translates of the homogeneous functions.

  • self similarity and multiwavelets in higher dimensions
    2004
    Co-Authors: Carlos Cabrelli, Christopher Heil, Ursula Molter
    Abstract:

    Introduction Matrices, tiles, and the joint spectral radius Generalized self-similarity and the Refinement Equation Multiresolution analysis Examples Bibliography Appendix A. Index of symbols.

  • Self-similarity and Multiwavelets in Higher Dimensions
    2004
    Co-Authors: Carlos Cabrelli, Christopher Heil, Ursula Molter
    Abstract:

    Introduction Matrices, tiles, and the joint spectral radius Generalized self-similarity and the Refinement Equation Multiresolution analysis Examples Bibliography Appendix A. Index of symbols.

  • Necessary conditions for the existence of multivariate multiscaling functions
    Wavelet Applications in Signal and Image Processing VIII, 2000
    Co-Authors: Carlos Cabrelli, Christopher Heil, Ursula Molter
    Abstract:

    In this paper we outline the main ideas behind the recent proof of the authors that if a multivariate, multi-function Refinement Equation with an arbitrary dilation matrix has a continuous, compactly supported solution which has independent lattice translates, then the joint spectral radius of certain matrices restricted to an appropriate subspace is strictly less than one.