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Let A be a d × d dilation matrix with integer entries and with determinant ± 2. The existence of Riesz wavelets with compact support associated to A is proved. Our proof is constructive and the generators of these Riesz wavelets may be taken to be symmetric, with high regularity and many vanishing moments. In our construction, we first study the structure of the quotient group ℤ d / A ℤ d . Afterwards and perhaps the main advance in this paper is the fact that we obtain a family of trigonometric polynomials on ℝ d with zeros only in (A ∗) − 1 ℤ d \ ℤ d . At this point, we are able to get scaling functions with compact support of a multiresolution analysis. In addition, Riesz wavelets with compact support hold by standard multiresolution techniques. Finally, we give some examples of Riesz wavelets where we emphasize on a numerical estimation of their regularity. [ABSTRACT FROM AUTHOR]

The regularity of refinable functions has been analysed in an extensive literature and is well-understood in two cases: 1) univariate 2) multivariate with an isotropic dilation matrix. The general (non-isotropic) case offered a great resistance. It was not before 2019 that the non-isotropic case was done by developing the matrix method. In this paper we make the next step and extend the Littlewood-Paley type method, which is very efficient in the aforementioned special cases, to general equations with arbitrary dilation matrices. This gives formulas for the higher order regularity in W 2 k (R n) by means of the Perron eigenvalue of a finite-dimensional linear operator on a special cone. Applying those results to recently introduced tile B-splines, we prove that they can have a higher smoothness than the classical ones of the same order. Moreover, the two-digit tile B-splines have the minimal support of the mask among all refinable functions of the same order of approximation. This proves, in particular, the lowest algorithmic complexity of the corresponding subdivision schemes. Examples and numerical results are provided. [ABSTRACT FROM AUTHOR]

For any dilation matrix with integral entries A ∈ R d × d , d ≥ 1 , we construct two families of Parseval wavelet frames in L 2 (R d). Both families have compact support and any desired number of vanishing moments. The first family has | det ⁡ A | generators. The second family has any desired degree of regularity. For the members of this family, the number of generators depends on the dilation matrix A and the dimension d , but never exceeds | det ⁡ A | + d. Our construction involves trigonometric polynomials developed by Heller to obtain refinable functions, the Oblique Extension Principle, and a slight generalization of a theorem of Lai and Stöckler. [ABSTRACT FROM AUTHOR]

Let A be an expansive linear map on R d preserving the integer lattice and with | det A | = 2 . We prove that if there exists a self-affine tile set associated to A, there exists a compactly supported wavelet with any desired number of vanishing moments and some symmetry. We put emphasis on construction of wavelets associated to a linear map A on R 2 and to the Quincunx dilation on R 3 because we can remove the hypothesis of the existence of the self-affine tile set. Our construction is based on low pass filters by Han in dimension one with the dyadic dilation and multiresolution theory. Finally, for some particular dilation matrices, we realize that unidimensional Daubechies low pass filers can be adapted to obtain compactly supported wavelets with any desired degree of regularity and any fix number of vanishing moments. [ABSTRACT FROM AUTHOR]

Construction of multivariate tight framelets is known to be a challenging problem because it is linked to the difficult problem on sum of squares of multivariate polynomials in real algebraic geometry. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either, since their construction is related to syzygy modules and factorization of multivariate polynomials. On the other hand, compactly supported multivariate framelets with directionality or high vanishing moments are of interest and importance in both theory and applications. In this paper we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. Let ϕ ∈ L 2 (R d) be an arbitrary compactly supported real-valued M -refinable function with a general dilation matrix M and ϕ ˆ (0) = 1 such that its underlying real-valued low-pass filter satisfies the basic sum rule. We first constructively prove by a step-by-step algorithm that we can always easily derive from the arbitrary M -refinable function ϕ a directional compactly supported real-valued quasi-tight M -framelet in L 2 (R d) associated with a directional quasi-tight M -framelet filter bank, each of whose high-pass filters has one vanishing moment and only two nonzero coefficients. If in addition all the coefficients of its low-pass filter are nonnegative, then such a quasi-tight M -framelet becomes a directional tight M -framelet in L 2 (R d). Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary M -refinable function ϕ a compactly supported quasi-tight M -framelet in L 2 (R d) with the highest possible order of vanishing moments. We shall also present a result on quasi-tight framelets whose associated high-pass filters are purely differencing filters with the highest order of vanishing moments. Several examples will be provided to illustrate our main theoretical results and algorithms in this paper. [ABSTRACT FROM AUTHOR]

Let d  ≥ 1. For any A ∈ Z d × d such that | det A | = 2 , we construct two families of Parseval wavelet frames with two generators. These generators have compact support, any desired number of vanishing moments, and any given degree of regularity. The first family is real valued while the second family is complex valued. To construct these families we use Daubechies low pass filters to obtain refinable functions, and adapt methods employed by Chui and He and Petukhov for dyadic dilations to this more general case. We also construct several families of Parseval wavelet frames with three generators having various symmetry properties. Our constructions are based on the same refinable functions and on techniques developed by Han and Mo and by Dong and Shen for the univariate case with dyadic dilations. [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]

In this paper we show some construction of nonseparable compactly supported bivariate wavelets. We deal with the dilation matrix \(A = \tiny{\left[\begin{matrix}0 & 2 \cr 1 & 0 \cr \end{matrix} \right]}\) and some three-row coefficient mask; that is a scaling function satisfies a dilation equation with scaling coefficients which can be contained in the set \(\{c_{n}\}_{n \in\mathcal{S}},\) where \(\mathcal{S}=S_{1} \times \{0,1,2\},\) \(S_{1} \subset \mathbb{N},\) \(\sharp S_{1} \lt \infty.\)

Let $$A \in \mathbb {R}^{d \times d}$$ , $$d \ge 1$$ be a dilation matrix with integer entries and $$| \det A|=2$$ . We construct several families of compactly supported Parseval framelets associated to A having any desired number of vanishing moments. The first family has a single generator and its construction is based on refinable functions associated to Daubechies low pass filters and a theorem of Bownik. For the construction of the second family we adapt methods employed by Chui and He and Petukhov for dyadic dilations to any dilation matrix A. The third family of Parseval framelets has the additional property that we can find members of that family having any desired degree of regularity. The number of generators is $$2^d+d$$ and its construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. For the particular case $$d=2$$ and based on the previous construction, we present two families of compactly supported Parseval framelets with any desired number of vanishing moments and degree of regularity. None of these framelet families have been obtained by means of tensor products of lower-dimensional functions. One of the families has only two generators, whereas the other family has only three generators. Some of the generators associated with these constructions are even and therefore symmetric. All have even absolute values. [ABSTRACT FROM AUTHOR]

In this paper we show the construction of nonseparable compactly supported bivariate wavelets. We deal with the dilation matrix A = [0/1 2/0] and some three-row coefficient 1 0 mask; that is a scaling function that satisfies a dilation equation with scaling coefficients which can be contained in the set {cn} nεS, where S = S1 x {0, 1, 2}, S1, ⊂, ℕ #1 < ∞. [ABSTRACT FROM AUTHOR]

We study multivariate trigonometric polynomials satisfying the 'sum-rule' conditions of a certain order. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest in the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. The approach presented in this paper leads directly to constructive algorithms, and is an alternative to the analysis of multivariate subdivision schemes in terms of the joint spectral radius of certain operators. Our convergence results apply to arbitrary dilation matrices, while the smoothness results are limited to two classes of dilation matrices. [ABSTRACT FROM AUTHOR]

We present a method for the construction of compactly supported ( 1 0 -1 1 1 0 1 0 1 ) wavelets under a mild condition. Wavelets inherit the symmetry of the corresponding scaling function and satisfies the vanishing moment condition originating in the symbols of the scaling function. As an application, an example is provided. [ABSTRACT FROM AUTHOR]

A refinable spline in ℝ d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, ), Lawton et al. (Comput. Math. 3, 137–145, ). In higher dimensions Sun (J. Approx. Theory 86, 240–252, ) characterized those splines when the dilation matrices are of the form A= mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ d for arbitrary dilation matrices A∈ M d(ℤ). [ABSTRACT FROM AUTHOR]

A refinable spline in ℝ d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, ), Lawton et al. (Comput. Math. 3, 137–145, ). In higher dimensions Sun (J. Approx. Theory 86, 240–252, ) characterized those splines when the dilation matrices are of the form A= mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ d for arbitrary dilation matrices A∈ M d(ℤ). [ABSTRACT FROM AUTHOR]

Abstract: Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are and dilation matrices with and . Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual -wavelet frames in are constructed, where , and with and both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual -wavelet frames, nonseparable -wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual -wavelet frames, nonseparable dual -wavelet frames with symmetry can be obtained easily. In the end, two examples are given. [Copyright &y& Elsevier]

In this paper, we study the convergence of non-orthogonal wavelet expansions associated withdilation matrix in the Lp(Rd) norm, 1 ≤p < ∞. Moreover, the results of Zayed [1] have been extended usingthe approximate identity but the present method and approach are different from those adopted by Zayed[1]. [ABSTRACT FROM AUTHOR]

Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d×d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight M-wavelet frames and orthonormal M-wavelet bases in L2(Rd) of exponential decay, which are derived from compactly supported M-refinable functions, such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments. This paper improves several results in Battle (Comm. Math. Phys. 110 (1987) 601), Bownik (J. Fourier Anal. Appl. 7(2001) 489), Gro¨chenig and Ron (Proc. Amer. Math. Soc. 126 (1998) 1101), Lemarie (J. Math. Pures Appl. 67 (1988) 227), and Strichartz (Constr. Approx. 9 (1993) 327). [Copyright &y& Elsevier]

In this note, we are interested in the symmetry property of a refinable function with a general dilation matrix. We investigate the symmetry group of a mask so that its associated refinable function with a general dilation matrix has a certain kind of symmetry. Given two dilation matrices which produce the same lattice, we demonstrate that if a mask has a certain kind of symmetry, then its associated refinable functions with respect to the two dilation matrices are the same; therefore, the two corresponding derived wavelet systems are essentially the same. Finally, we illustrate that for any dilation matrix, orthogonal masks, as well as interpolatory masks having nonnegative symbols, can be easily constructed with any preassigned order of sum rules by employing a linear transform. Without solving any equations, the method in this note on constructing masks with certain desirable properties is simple, painless and general. Examples of quincunx wavelets and wavelets with respect to the checkerboard lattice are presented to illustrate the general theory. [Copyright &y& Elsevier]

A biorthogonal wavelet is derived from a pair of biorthogonal refinable functions using the standard technique in multiresolution analysis. In this paper, we introduce the concept of projectable refinable functions and demonstrate that many multivariate refinable functions are projectable; that is, they essentially carry the tensor product (separable) structure though themselves may be nontensor product (nonseparable) refinable functions. For any pair of biorthogonal refinable functions (φ,φd) in L2(Rs), when the refinable function φ is projectable, we prove that without loss of several desirable properties such as spatial localization, smoothness and approximation order, from the pair of biorthogonal refinable functions (φ,φd), we can easily obtain another pair of biorthogonal refinable functions in L2(Rs) which are tensor product separable refinable functions. As an application, we show that there is no dual refinable function φd to the refinable basis function in the Loop scheme such that φd can be supported on [−4,4]2. [Copyright &y& Elsevier]

Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σk∈Λ ck f(Ax−k), where Λ is a finite subset of Γ, the ck are r×r matrices, and f=(f1,...,fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)

1,...,fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα=ΣΣk∈Γyα,i(k) fi(x+k). These coefficients are multivariate polynomials yα,i(x) of degree |α| evaluated at lattice points k∈Γ. [ABSTRACT FROM AUTHOR]

This letter derives a family of dilation matrices for the body-centered cubic (BCC) lattice, which is optimal in the sense of spectral sphere packing. While satisfying the necessary conditions for dilation, these matrices are all cube roots of an integer scalar matrix. This property offers theoretical advantages for construction of wavelet functions in addition to the practical advantages when iterating through a perfect reconstruction filter bank based on BCC downsampling. Lastly, we factor the BCC matrix into two matrices that allow us to cascade two two-channel perfect reconstruction filter banks in order to construct a four-channel perfect reconstruction filter bank based on BCC downsampling. [ABSTRACT FROM AUTHOR]