Construction of wavelet sets with certain self-similarity properties

A measurable set Q ⊂ R ⁿ is a wavelet set for an expansive matrix A if F ⁻¹ _(ΧQ) is an A-dilation wavelet. Dai, Larson, and Speegle [7] discovered the existence of wavelet sets in R _n associated with any real n ×n expansive matrix. In this work, we construct a class of compact wavelet sets which do not contain the origin and which are, up to a certain linear transformation, finite unions of integer translates of an integral selfaffine tile associated with the matrix B = A ^t. Some of these wavelet sets may have good potential for applications because of their tractable geometric shapes.     

On the sampling theorem for wavelet subspaces

In [13], Walter extended the classical Shannon sampling theorem to some wavelet subspaces. For any closed subspace V₀/L² (R), we present a necessary and sufficient condition under which there is a sampling expansion for everyf V₀-Several examples are given.     

Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems

In this paper we classify the centers localized at the origin of coordinates, the cyclicity of their Hopf bifurcation and their isochronicity for the polynomial differential systems in \mathbbR²{\mathbb{R}^2} of degree d that in complex notation z = x + i y can be written as

A characterization of wavelets on general lattices

In the context of a general lattice Γ in Rⁿ and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families. This result generalizes the characterization of Frazier, Garrigós, Wang, and Weis about the wavelet families with Γ = Zⁿ and M = 21. In the second part of the paper, we characterize all the MSF wavelets. Moreover, we give a constructive method for the support of the Fourier transform of an MSF wavelet and apply this method by giving examples with particular attention to the quincunx lattice.     

ADMISSIBLE WAVELETS ASSOCIATED WITH THE TRANSFORM GROUP ON R

Let p be the transform group on R, then P has a natural unitary representation U onL2 (R^n). Decompose L2(R^n) into the direct sum of irreducible invariant closed subspace,s. The re-striction of U on these suhspaces is square-intagrable. In this paper the characterization of admissi-ble condition in tarrns of the Fourier transform is given. The wavelet transform is defined, and theorthogorml direct sum decomposition of function space L2 (P,du1) is obtained.     

Embedding a polytope in a lattice

We present a special similarity ofR ⁴ⁿ which maps lattice points into lattice points. Applying this similarity, we prove that if a (4n−1)-polytope is similar to a lattice polytope (a polytope whose vertices are all lattice points) inR ⁴ⁿ , then it is similar to a lattice polytope inR ⁴ⁿ⁻¹, generalizing a result of Schoenberg [4]. We also prove that ann-polytope is similar to a lattice polytope in someR ^N if and only if it is similar to a lattice polytope inR ²ⁿ⁺¹, and if and only if sin²(<ABC) is rational for any three verticesA, B, C of the polytope.     

Density of weighted wavelet frames

If ψ ∈ L²(R), Λ is a discrete subset of the affine groupA =R ⁺ ×R, and w: Λ →R ⁺ is a weight function, then the weighted wavelet system generated by ψ, Λ, and w is . In this article we define lower and upper weighted densities D _w ⁻ (Λ) and D _w ⁺ (Λ) of Λ with respect to the geometry of the affine group, and prove that there exist necessary conditions on a weighted wavelet system in order that it possesses frame bounds. Specifically, we prove that if W(ψ, Λ, w) possesses an upper frame bound, then the upper weighted density is finite. Furthermore, for the unweighted case w = 1, we prove that if W(ψ, Λ, 1) possesses a lower frame bound and D _w ⁺ (Λ⁻¹) < ∞, then the lower density is strictly positive. We apply these results to oversampled affine systems (which include the classical affine and the quasi-affine systems as special cases), to co-affine wavelet systems, and to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems.     

Symmetric orthogonal filters and wavelets with linear-phase moments

In this paper we study symmetric orthogonal filters with linear-phase moments, which are of interest in wavelet analysis and its applications. We investigate relations and connections among the linear-phase moments, sum rules, and symmetry of an orthogonal filter. As one of the results, we show that if a real-valued orthogonal filter a is symmetric about a point, then a has sum rules of order m if and only if it has linear-phase moments of order 2m. These connections among the linear-phase moments, sum rules, and symmetry help us to reduce the computational complexity of constructing symmetric real-valued orthogonal filters, and to understand better symmetric complex-valued orthogonal filters with linear-phase moments. To illustrate the results in the paper, we provide many examples of univariate symmetric orthogonal filters with linear-phase moments. In particular, we obtain an example of symmetric real-valued 4-orthogonal filters whose associated orthogonal 4-refinable function lies in C²(R).     

Regularity of refinable function vectors

We study the existence and regularity of compactly supported solutions φ = (φ_v) _v=0 ^/r−1 of vector refinement equations. The space spanned by the translates of φ_v can only provide approximation order if the refinement maskP has certain particular factorization properties. We show, how the factorization ofP can lead to decay of |̸_v(u)| as |u| → ∞. The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm.     

Uniqueness theorem of solutions for stochastic differential equation in the plane

LetM={M _z, z ∈ R ₊ ² } be a continuous square integrable martingale andA={A _z, z ∈ R ₊ ² be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane:dX _z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R ₊ ² , X_z=Z_z, z∈∂R ₊ ² , whereR ₊ ² =[0, +∞)×[0,+∞) and ∂R ₊ ² is its boundary,Z is a continuous stochastic process on ∂R ₊ ² . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. Supported by the National Science Foundation and the Postdoctoral Science Foundation of China