In this paper we present the construction of new stable biorthogonal spline-wavelet bases on the interval [0, 1] for arbitrary choice of spline-degree. As starting point, we choose the well-known family of compactly supported biorthogonal spline-wavelets presented by Cohen, Daubechies and Feauveau. Firstly, we construct biorthogonal MRAs (multiresolution analysis) on [0, 1]. The primal MRA consists of spline-spaces concerning equidistant, dyadic partitions of [0, 1], the so called Schoenberg-spline bases. Thus, the full degree of polynomial reproduction is preserved on the primal side. The construction, that we present for the boundary scaling functions on the dual side, guarantees the same for the dual side. In particular, the new boundary scaling functions on both, the primal and the dual side have staggered supports. Further, the MRA spaces satisfy certain Jackson-and Bernstein-inequalities, which lead by general principles to the result, that the associated wavelets are in fact L2([0, 1])-stable. The wavelets however are computed with aid of the method of stable completion. Due to the compact support of all occurring functions, the decomposition and reconstruction transforms can be implemented efficiently with sparse matrices. We also illustrate how bases with complementary or homogeneous boundary conditions can be easily derived from our construction.
Mathematics Subject Classification (2000). Primary 42C40; Secondary 65F30. Keywords. Multiresolution on the interval, biorthogonal spline wavelets, Riesz bases, characterization of Sobolev spaces, boundary conditions. 122 M. Primbs Results. Math.
In this paper, we discuss a simple method for the numerical computation of Gramian matrices, that appears within the construction of multiresolution analysis on the interval. The presented approach covers all, the orthogonal, the semiorthogonal and the biorthogonal cases. We consider constructions, which are based on translates of a known pair of compactly supported functions ϕ,[Formula: see text] inside the interval, and supplemented by boundary functions. Using the given two-scale-coefficients of these boundary functions, we can reduce the problem to a linear system of equations. Furthermore, we discuss conditions providing that these systems are uniquely solvable. In particular, no integrals have to be computed numerically. Finally, using the Schoenberg spline basis on the interval as an example, we show how to apply the method to a well-known problem.
This article is concerned with adaptive numerical frame methods for elliptic operator equations. We show how specific noncanonical frame expansions on domains can be constructed. Moreover, we study the approximation order of best n-term frame approximation, which serves as the benchmark for the performance of adaptive schemes. We also discuss numerical experiments for second order elliptic boundary value problems in polygonal domains where the discretization is based on recent constructions of boundary adapted wavelet bases on the interval.
In this paper we examine, how the method of stable completion presented in [1] fits into the classical framework of wavelet construction as known from [2,6]. We focus on the biorthogonal case (which includes the orthogonal one) as considered in [3]. We show, that using the initial stable completion as constructed in [5], the resulting wavelets coincide up to scaling and shifting with the original wavelets from [3]. The scaling factor is specified.
Mathematics Subject Classification (2000). Primary 42C40; Secondary 65F30, 15-99.
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