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When constructing multivariate framelets, it is often unavoidable to work with matrices of multivariate trigonometric polynomials and complicated matrix decomposition problems. These problems become even harder when good properties such as high-order vanishing moments are required on the framelets. In this paper, we establish a new method for constructing multivariate dual framelets with high-order vanishing moments. The underlying scheme of our algorithm is the famous Mixed Extension Principle that allows us to derive the high-pass filters (or framelet generators) from a given pair of refinable filters with high-order linear-phase moments. Our method only involves two steps: (1) directly constructing the first few pairs of high-pass filters by using the linear-phase moment conditions of the refinement filters; (2) solving a system of linear equations to obtain the rest of the high-pass filters. Both are easy to implement for scientific computation, regardless of what dimension or dilation matrix we work with. Apart from high-order vanishing moments, we will see that if the refinement filters take coefficients from some subfield of ℂ that is closed under complex conjugation, so do the high-pass filters. Furthermore, our algorithm gives the upper bounds for the number of high-pass filters in arbitrary dimensions. At the end of the paper, we will give several illustrative examples, from which we can also see that the support sizes of the high-pass filters are comparable with those of the refinement filters. [ABSTRACT FROM AUTHOR]

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

For any symmetry group , any appropriate matrix dilation (compatible with ) and any appropriate symmetry center we give an explicit method for the construction of -symmetric with respect to the center refinable masks which have sum rule of an arbitrary order . Moreover, we give a description of all these masks. For any symmetry group , any appropriate matrix dilation (compatible with ) and any appropriate row of symmetry centers we give two explicit methods for the construction of -symmetric with respect to the row of centers refinable matrix masks which have sum rule of an arbitrary order . A description of all such matrix masks is also presented. [ABSTRACT FROM AUTHOR]

Research on refinable functions in wavelet theory is mostly focused to localized functions. However it is known, that polynomial functions are refinable, too. In our paper we investigate on conversions between refinement masks and polynomials and their uniqueness. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for constructing a class of orthogonal two-direction refinable functions and the corresponding orthogonal two-direction wavelets is obtained. In addition, the relation of both two-direction refinable functions and multiwavelets is discussed. We discover that Chui–Lian's orthogonal symmetric/antisymmetric multiscaling functions and the corresponding multiwavelets can be recovered by using an orthogonal two-direction refinable function and the corresponding two-direction wavelets, respectively. Finally, some construction examples are given. [ABSTRACT FROM AUTHOR]

Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form \[ \varphi(x) = \sum_{k\in\Lambda} c_k \varphi(Ax - k),\quad c_k\in {\mathbb C}, \] 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 $\mathcal{H}$ in the shift-invariant space generated by φ. We provide a basis of $\mathcal{H}$ 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 $\mathcal{H}$ 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. [ABSTRACT FROM AUTHOR]

Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form \[ \varphi(x) = \sum_{k\in\Lambda} c_k \varphi(Ax - k),\quad c_k\in {\mathbb C}, \] 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 $\mathcal{H}$ in the shift-invariant space generated by φ. We provide a basis of $\mathcal{H}$ 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 $\mathcal{H}$ 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. [ABSTRACT FROM AUTHOR]

We present a technique for studying refinable functions which are compactly supported. Refinable functions satisfy dilation equations and this technique focuses on the implications of the dilation equation at the edges of the support of the refinable function. This method is fruitful, producing new results regarding existence, uniqueness, smoothness and rate of growth of refinable functions. [ABSTRACT FROM AUTHOR]