Smoothness of refinable function vectors on ℝn

Ramazan Tinaztepe; Denise Jacobs; Christopher Heil

doi:10.1142/s0219691317500515

Smoothness of refinable function vectors on ℝn

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691317500515 ◽

2017 ◽

Vol 15 (05) ◽

pp. 1750051

Author(s):

Ramazan Tinaztepe ◽

Denise Jacobs ◽

Christopher Heil

Keyword(s):

Scaling Function ◽

Joint Spectral Radius ◽

Refinement Equation ◽

Dilation Matrix ◽

Finite Set ◽

Expansive Matrix ◽

Refinable Function Vectors ◽

Multiplicity Formula ◽

Compactly Supported Solutions ◽

Order Smoothness

Let [Formula: see text] be a dilation matrix, an [Formula: see text] expansive matrix that maps [Formula: see text] into itself. Let [Formula: see text] be a finite subset of [Formula: see text] and for [Formula: see text] let [Formula: see text] be [Formula: see text] complex matrices. The refinement equation corresponding to [Formula: see text] and [Formula: see text] is [Formula: see text] A solution [Formula: see text] if one exists, is called a refinable vector function or a vector scaling function of multiplicity [Formula: see text] This paper characterizes the higher-order smoothness of compactly supported solutions of the refinement equation, in terms of the [Formula: see text]-norm joint spectral radius of a finite set of finite matrices determined by the coefficients [Formula: see text]

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Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

Journal of Applied Mathematics ◽

10.1155/2013/896050 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Fengjuan Zhu ◽

Qiufu Li ◽

Yongdong Huang

Keyword(s):

Scaling Function ◽

Minimum Energy ◽

Sufficient Conditions ◽

Reconstruction Algorithm ◽

Wavelet Function ◽

Necessary Condition ◽

Wavelet Frame ◽

Wavelet Frames ◽

Numerical Examples ◽

Dilation Matrix

In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of the minimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept of minimum-energy bivariate wavelet frame is defined, and its equivalent characterizations and a necessary condition are presented. Secondly, based on polyphase form of symbol functions of scaling function and wavelet function, two sufficient conditions and an explicit constructed method are given. Finally, the decomposition algorithm, reconstruction algorithm, and numerical examples are designed.

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REFINABLE SHIFT INVARIANT SPACES IN ℝd

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691305000889 ◽

2005 ◽

Vol 03 (03) ◽

pp. 321-345

Author(s):

CARLOS A. CABRELLI ◽

SIGRID B. HEINEKEN ◽

URSULA M. MOLTER

Keyword(s):

Dimensional Subspace ◽

Finite Dimensional Subspace ◽

Invariant Space ◽

Refinement Equation ◽

Compactly Supported Function ◽

Dilation Matrix ◽

Jordan Basis ◽

Shift Invariant Spaces ◽

Invariant Spaces ◽

Shift Invariant Space

Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form [Formula: see text] 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 [Formula: see text] in the shift-invariant space generated by φ. We provide a basis of [Formula: see text] 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 [Formula: see text] 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.

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Design of High Dimensional Nonseparable Compactly Supported Wavelets with Special Dilation Matrix

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.542-543.547 ◽

2012 ◽

Vol 542-543 ◽

pp. 547-550

Author(s):

Lan Li

Keyword(s):

Mild Condition ◽

Scaling Function ◽

New Method ◽

High Dimensional ◽

Moment Condition ◽

Compactly Supported ◽

Vanishing Moment ◽

Dilation Matrix ◽

The Scaling Function

In this paper, a new method to construct the compactly supported M- wavelet under a mild condition are given. The constructed wavelet satisfies the vanishing moment condition which is originated from the symbols of the scaling function.

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Extremal sequences of polynomial complexity

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004113000157 ◽

2013 ◽

Vol 155 (2) ◽

pp. 191-205 ◽

Cited By ~ 3

Author(s):

KEVIN G. HARE ◽

IAN D. MORRIS ◽

NIKITA SIDOROV

Keyword(s):

Polynomial Complexity ◽

Exponential Growth Rate ◽

Maximal Rate ◽

Joint Spectral Radius ◽

Bounded Set ◽

Fixed Set ◽

Finite Set ◽

Dimension 2 ◽

Subword Complexity ◽

Real Matrices

AbstractThe joint spectral radius of a bounded set of d × d real matrices is defined to be the maximum possible exponential growth rate of products of matrices drawn from that set. For a fixed set of matrices, a sequence of matrices drawn from that set is called extremal if the associated sequence of partial products achieves this maximal rate of growth. An influential conjecture of J. Lagarias and Y. Wang asked whether every finite set of matrices admits an extremal sequence which is periodic. This is equivalent to the assertion that every finite set of matrices admits an extremal sequence with bounded subword complexity. Counterexamples were subsequently constructed which have the property that every extremal sequence has at least linear subword complexity. In this paper we extend this result to show that for each integer p ≥ 1, there exists a pair of square matrices of dimension 2p(2p+1 − 1) for which every extremal sequence has subword complexity at least 2−p2np.

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Construction of a Class of Nonseparable Compactly Supported Wavelets with Special Dilation Matrix in L2(R4)

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.393-395.659 ◽

2011 ◽

Vol 393-395 ◽

pp. 659-662

Author(s):

Na Li

Keyword(s):

Mild Condition ◽

Scaling Function ◽

Moment Condition ◽

Compactly Supported ◽

Vanishing Moment ◽

Dilation Matrix ◽

Novel Method ◽

The Scaling Function

In this paper, a novel method to construct the compactly supported wavelet under a mild condition. The constructed wavelet satisfies the vanishing moment condition which is originated from the symbols of the scaling function.

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Polynomial spaces reproduced by elliptic scaling functions

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691315500423 ◽

2015 ◽

Vol 13 (06) ◽

pp. 1550042 ◽

Cited By ~ 2

Author(s):

Victor G. Zakharov

Keyword(s):

Differential Operator ◽

Quadratic Form ◽

Scaling Function ◽

Polynomial Solution ◽

Invariant Polynomial ◽

Scaling Functions ◽

Polynomial Space ◽

Scale Invariant ◽

Compactly Supported ◽

Dilation Matrix

In this paper, we consider the so-called elliptic scaling functions [V. G. Zakharov, Elliptic scaling functions as compactly supported multivariate analogs of the B-splines, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014) 1450018]. Any elliptic scaling function satisfies the refinement relation with a real isotropic dilation matrix; and, in the paper, we prove that any real isotropic matrix is similar to an orthogonal matrix and the similarity transformation matrix determines a positive-definite quadratic form. We prove that the polynomial space reproduced by integer shifts of a compactly supported function can be usually considered as a polynomial solution to a system of constant coefficient PDE’s. We show that the algebraic polynomials reproduced by a compactly supported elliptic scaling function belong to the kernel of a homogeneous elliptic differential operator that the differential operator corresponds to the quadratic form; and thus any elliptic scaling function reproduces only affinely-invariant polynomial spaces. However, in the paper, we present nonstationary elliptic scaling functions such that the scaling functions can reproduce no scale-invariant (only shift-invariant) polynomial spaces.

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Characterization of Generators for Multiresolution Analyses with Composite Dilations

Abstract and Applied Analysis ◽

10.1155/2011/850850 ◽

2011 ◽

Vol 2011 ◽

pp. 1-13

Author(s):

Yuan Zhu ◽

Wenjun Gao ◽

Dengfeng Li

Keyword(s):

General Theory ◽

Scaling Function ◽

Reducing Subspace ◽

Reducing Subspaces ◽

Expansive Matrix

This paper introduces multiresolution analyses with composite dilations (AB-MRAs) and addresses frame multiresolution analyses with composite dilations in the setting of reducing subspaces ofL2(ℝn)(AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspaceL2(S)∨. For a general expansive matrix, we obtain the characterizations for a scaling function to generate an AB-RMRA, and the main theorems generalize the classical results. Finally, some examples are provided to illustrate the general theory.

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