The Experts below are selected from a list of 6 Experts worldwide ranked by ideXlab platform
Bin Han - One of the best experts on this subject based on the ideXlab platform.
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Symmetric Orthonormal Scaling Functions and wavelets with dilation factor 4
Advances in Computational Mathematics, 1998Co-Authors: Bin HanAbstract:It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric Orthonormal Scaling Function except for the Haar Function. In this paper we are concerned with the construction of symmetric Orthonormal Scaling Functions with dilation factor d=4. Several examples of such Orthonormal Scaling Functions are provided in this paper. In particular, two examples of C1 Orthonormal Scaling Functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how to construct symmetric wavelets from these Scaling Functions. We explicitly construct the corresponding Orthonormal symmetric wavelets for all the examples given in this paper.
Patrick J. Van Fleet - One of the best experts on this subject based on the ideXlab platform.
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gibbs phenomenon for nonnegative compactly supported Scaling vectors
Journal of Mathematical Analysis and Applications, 2005Co-Authors: David K Ruch, Patrick J. Van FleetAbstract:Abstract This paper considers Gibbs' phenomenon for Scaling vectors in L 2 ( R ) . We first show that a wide class of multiresolution analyses suffer from Gibbs' phenomenon. To deal with this problem, in [Contemp. Math. 216 (1998) 63–79], Walter and Shen use an Abel summation technique to construct a positive Scaling Function P r , 0 r 1 , from an Orthonormal Scaling Function ϕ that generates V 0 . A reproducing kernel can in turn be constructed using P r . This kernel is also positive, has unit integral, and approximations utilizing it display no Gibbs' phenomenon. These results were extended to Scaling vectors and multiwavelets in [Proceedings of Wavelet Analysis and Multiresolution Methods, 2000, pp. 317–339]. In both cases, orthogonality and compact support were lost in the construction process. In this paper we modify the approach given in [Proceedings of Wavelet Analysis and Multiresolution Methods, 2000, pp. 317–339] to construct compactly supported positive Scaling vectors. While the mapping into V 0 associated with this new positive Scaling vector is not a projection, the Scaling vector does produce a Riesz basis for V 0 and we conclude the paper by illustrating that expansions of Functions via positive Scaling vectors exhibit no Gibbs' phenomenon.
David K Ruch - One of the best experts on this subject based on the ideXlab platform.
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gibbs phenomenon for nonnegative compactly supported Scaling vectors
Journal of Mathematical Analysis and Applications, 2005Co-Authors: David K Ruch, Patrick J. Van FleetAbstract:Abstract This paper considers Gibbs' phenomenon for Scaling vectors in L 2 ( R ) . We first show that a wide class of multiresolution analyses suffer from Gibbs' phenomenon. To deal with this problem, in [Contemp. Math. 216 (1998) 63–79], Walter and Shen use an Abel summation technique to construct a positive Scaling Function P r , 0 r 1 , from an Orthonormal Scaling Function ϕ that generates V 0 . A reproducing kernel can in turn be constructed using P r . This kernel is also positive, has unit integral, and approximations utilizing it display no Gibbs' phenomenon. These results were extended to Scaling vectors and multiwavelets in [Proceedings of Wavelet Analysis and Multiresolution Methods, 2000, pp. 317–339]. In both cases, orthogonality and compact support were lost in the construction process. In this paper we modify the approach given in [Proceedings of Wavelet Analysis and Multiresolution Methods, 2000, pp. 317–339] to construct compactly supported positive Scaling vectors. While the mapping into V 0 associated with this new positive Scaling vector is not a projection, the Scaling vector does produce a Riesz basis for V 0 and we conclude the paper by illustrating that expansions of Functions via positive Scaling vectors exhibit no Gibbs' phenomenon.
