Topic
Biorthogonal system
About: Biorthogonal system is a research topic. Over the lifetime, 2190 publications have been published within this topic receiving 32209 citations.
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01 Jan 1994
TL;DR: In this paper, a Galerkin approach was used for the solution of an elliptic boundary value problem in one dimension with polynomial coefficients, where biorthogonal wavelets adapted to a differential operator with constant coefficients were used to set up the system of linear equations.
Abstract: We will be concerned with the solution of an elliptic boundary, value problem in one dimension with polynomial coefficients. In a Galerkin approach, we employ biorthogonal wavelets adapted to a differential operator with constant coefficients, and use the refinement equations to set up the system of linear equations with exact entries (up to round-off). For the solution of the linear equation, we construct a biorthogonal two-grid method with intergrid operators stemming from wavelet-type operators adapted to the problem.
21 citations
TL;DR: Two types of vector wavelet transforms are described that are dual to each other but have similar properties and three pre(post)processing algorithms that do not degrade the approximation accuracy of the vector wavelets.
Abstract: We present new multiwavelet transforms of multiplicity 2 for manipulation of discrete-time signals. The transforms are implemented in two phases: (1) pre(post)-processing, which transforms the scalar signal into a vector signal (and back) and (2) wavelet transforms of the vector signal. Both phases are performed in a lifting manner. We use the cubic interpolatory Hermite splines as a predicting aggregate in the vector wavelet transform. We present new pre(post)-processing algorithms that do not degrade the approximation accuracy of the vector wavelet transforms. We describe two types of vector wavelet transforms that are dual to each other but have similar properties and three pre(post)processing algorithms. As a result, we get fast biorthogonal algorithms to transform discrete-time signals that are exact on sampled cubic polynomials. The bases for the transform are symmetric and have short support.
21 citations
TL;DR: This work proves optimal a priori estimates for both stream function and vorticity, and presents numerical results to demonstrate the efficiency of the approach.
Abstract: We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optimal a priori estimates for both stream function and vorticity, and present numerical results to demonstrate the efficiency of the approach.
21 citations
TL;DR: In this article, it was shown that the relevant biorthogonal polynomials are actually special cases of multiple orthogonal ensembles associated with modified Bessel functions, which was first introduced by Van Assche and Yakubovich.
Abstract: The product of M complex random Gaussian matrices of size N has recently been studied by Akemann, Kieburg, and Wei. They showed that, for fixed M and N, the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed. We find that, in the case M = 2, the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with Macdonald functions (modified Bessel functions of the second kind) which was first introduced by Van Assche and Yakubovich. With known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles, it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random Gaussian matrices in the limit of large matrix dimensions.
21 citations
04 Oct 1998
TL;DR: This paper presents shape adaptive wavelet transforms for object-based image coding and methods for recovering moment properties that are valid for the original filters and wavelets, but that get lost in the boundary regions of shape adaptive transforms.
Abstract: This paper presents shape adaptive wavelet transforms for object-based image coding. Methods for recovering moment properties that are valid for the original filters and wavelets, but that get lost in the boundary regions of shape adaptive transforms, are presented. Furthermore, methods for equalizing the energies of the boundary wavelets are shown. This equalization allows one to avoid the white quantization noise (introduced in the subbands) that appears as highly colored noise in the reconstructed image.
21 citations