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Abstract: The purpose of this paper is to investigate explicit iteration schemes for minimization problems arising from image denoising. In particular, we propose explicit iteration schemes based on matrix splitting. When the matrix splitting is done by the symmetric Gauss–Seidel method, we establish convergence of the scheme with no restriction on the step size of the iteration. If the matrix splitting is done by the Gauss–Seidel method, we show that the iteration scheme still converges, provided the step size of each iteration is sufficiently small. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate approximation of quasi-projection operators in Besov spaces , , . Suppose is a countable index set. Let be a family of functions in , and let be a family of functions in , where . Let be the quasi-projection operator given by For , by we denote the scaling operator given by , . Let . Under some mild conditions on the functions and (), we establish the following result: If , and if for all polynomials of degree at most , then the estimate is valid for all , where is a constant independent of and . Density of quasi-projection operators in Besov spaces is also discussed. [Copyright &y& Elsevier]

Abstract: The total variation model of Rudin, Osher, and Fatemi for image denoising is considered to be one of the best denoising models. Recently, by using the Bregman method, Goldstein and Osher obtained a very efficient algorithm for the solution of the ROF model. In this paper, we give a rigorous proof for the convergence of the Bregman method. We also indicate that a combination of the Bregman method with wavelet packet decomposition often enhances performance for certain texture rich images. [Copyright &y& Elsevier]

Abstract: We investigate Riesz bases of wavelets generated from multiresolution analysis. This investigation leads us to a study of refinement equations with masks being exponentially decaying sequences. In order to study such refinement equations we introduce the cascade operator and the transition operator. It turns out that the transition operator associated with an exponentially decaying mask is a compact operator on a certain Banach space of sequences. With the help of the spectral theory of the compact operator we are able to characterize the convergence of the cascade algorithm associated with an exponentially decaying mask in terms of the spectrum of the corresponding transition operator. As an application of this study we establish the main result of this paper which gives a complete characterization of all possible Riesz bases of compactly supported wavelets generated from multiresolution analysis. Several interesting examples are provided to illustrate the general theory. [Copyright &y& Elsevier]

Abstract: In this paper we investigate Bessel sequences in the space , in Sobolev spaces (), and in Besov spaces (). For each , let be a countable index set. Let be a family of functions in . We give some sufficient conditions for the family to be a Bessel sequence in or . The results obtained in this paper are useful for the study of frames and Riesz bases for or . In particular, these results are applicable to wavelets on irregular meshes. [Copyright &y& Elsevier]

Abstract: The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis. [Copyright &y& Elsevier]

In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and φ˜ in L2(R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψjk:=2j/2ψ(2j·−k) (j,k∈Z) form a Riesz basis for L2(R). If, in addition, φ lies in the Sobolev space Hm(R), then the derivatives 2j/2ψ(m)(2j·−k) (j,k∈Z) also form a Riesz basis for L2(R). Consequently, {ψjk: j,k∈Z} is a stable wavelet basis for the Sobolev space Hm(R). The pair of φ and φ˜ are not required to be biorthogonal or semi-orthogonal. In particular, φ and φ˜ can be a pair of B-splines. The added flexibility on φ and φ˜ allows us to construct wavelets with relatively small supports. [Copyright &y& Elsevier]

In this paper we consider functional equations of the form Φ=∑lower limit α∈Zs a(α)Φ(M·−α), where Φ=(φ1,…,φr)T is an r×1 vector of functions on the s-dimensional Euclidean space, a(α), α∈Zs, is a finitely supported sequence of r×r complex matrices, and M is an s×s isotropic integer matrix such that lim n→∞M−n=0. We are interested in the question, for which sequences a will there exist a solution to the functional equation with each function φj, j=1,…,r, belonging to the Sobolev space Wkp(Rs)? Our approach will be to consider the convergence of the cascade algorithm. The cascade operator Qa associated with the sequence a is defined by QaF:=∑lower limit α∈Zs a(α)F(M·−α), F∈(Wkp(Rs))r. Let Φ0 be a nontrivial r×1 vector of compactly supported functions in Wkp(Rs). The iteration scheme Φn=QaΦn−1, n=1,2,… , is called a cascade algorithm, or a subdivision scheme. Under natural assumptions on a, a feasible set of initial vectors is identified from the conditions on an initial vector implied by the convergence of the subdivision scheme. These conditions are determined by the matrix A(0)=m−1∑α∈Zsa(α), m=∣det M∣, and are related to polynomial reproducibility and the classical Strang–Fix conditions.The formal definition of convergence in the Sobolev norm for the subdivision scheme is that the scheme will converge for any choice of initial vector from the feasible set (to the same solution Φ). We give a characterization for this concept of convergence in terms of the p-norm joint spectral radius of a finite collection of transition operators determined by the sequence a restricted to a certain invariant subspace. The invariant subspace is intimately connected to the Strang–Fix type conditions that determine the feasible set of initial vectors. [Copyright &y& Elsevier]