拟样条的几点研究
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摘要
为了构造高逼近阶的紧小波框架,Daubechies,Han,Ron和Shen引入了一型拟样条.与此同时,Selesnick也独立地介绍了一型拟样条.从考虑对称性的角度出发,Dong和Shen构造了二型拟样条.拟样条提供了很多新的细分函数.它包含了插值细分函数,正交细分函数和B样条作为特例.Han和Shen利用拟样条的面具构造了一系列紧支集C~∞的非固定细分函数并发展了一套与之对应的非固定紧小波框架的理论.本文研究了拟样条的几个性质,主要从以下几个方面展开;
·拟样条的正则性;
·构造对称复拟样条,通过复拟样条,我们构造L_2(R)中具有高逼近阶的对称或逆对称紧小波框架;
·结合拟样条和箱样条这两个定义,我们定义了拟箱样条,拟箱样条是拟样条从一维到高维的一个自然的推广;
·通过拟样条的面具,构造对称C~∞非稳定紧支集复拟样条和拟箱样条.
另外我们研究了无限支集面具在L_2(R)空间中的细分算法收敛性.同时,我们证明了对于一类无限支集具有H(O|¨)lder连续性的面具,我们都能构造无限支集的Riesz小波基.对于给定的非齐次细分方程,我们给出一个一般算法,通过这个算法,我们对非齐次细分向量的每个分量的支集都做出了精确的估计.
The first type of pseudo splines were introduced by Daubechies,Han,Ron and Shen to construct tight wavelets frames with high approximation order.They were introduced by Selesnick independently.Consider symmetry,Dong and Shen constructed the second type of pseudo splines.Pseudo splines provides a rich family of refinable function.B-splines,orthogonal refinable functions and interpolation refinable functions are special cases of pseudo splines.With the masks of pseudo splines,Han and Shen constructed C~∞nonstationary refinable functions with compact support and give a complete analysis of nonstationary tight wavelets frames. In this paper,we investigate some properties of pseudo splines as following,
·Give a regularity analysis of pseudo splines;
·Construct symmetric complex pseudo splines and symmetric or antisymmetric complex tight wavelets frames with high approximation order in L_2(R);
·Combine the definitions of B-splines and Box splines,we define pseudo box splines.Pseudo box splines are one multivariate generalization of univariate pseudo splines.
·With the masks of pseudo splines,we construct nonstationary symmetric C~∞complex pseudo splines and pseudo box splines.
Moreover,we investigate the convergence of cascade algorithm associated with the masks h(o|¨)lder continuous.We construct Riesz wavelet bases in L_2(R) with fast decay.We also investigate the support of a refinable vector satisfying an inhomogeneous refinement equation.By using some methods introduced by So and Wang, an estimate is given for the support of each component function of a compactly supported refinable vector satisfying an inhomogeneous matrix refinement equation with finitely supported masks.
·拟样条的正则性;
·构造对称复拟样条,通过复拟样条,我们构造L_2(R)中具有高逼近阶的对称或逆对称紧小波框架;
·结合拟样条和箱样条这两个定义,我们定义了拟箱样条,拟箱样条是拟样条从一维到高维的一个自然的推广;
·通过拟样条的面具,构造对称C~∞非稳定紧支集复拟样条和拟箱样条.
另外我们研究了无限支集面具在L_2(R)空间中的细分算法收敛性.同时,我们证明了对于一类无限支集具有H(O|¨)lder连续性的面具,我们都能构造无限支集的Riesz小波基.对于给定的非齐次细分方程,我们给出一个一般算法,通过这个算法,我们对非齐次细分向量的每个分量的支集都做出了精确的估计.
The first type of pseudo splines were introduced by Daubechies,Han,Ron and Shen to construct tight wavelets frames with high approximation order.They were introduced by Selesnick independently.Consider symmetry,Dong and Shen constructed the second type of pseudo splines.Pseudo splines provides a rich family of refinable function.B-splines,orthogonal refinable functions and interpolation refinable functions are special cases of pseudo splines.With the masks of pseudo splines,Han and Shen constructed C~∞nonstationary refinable functions with compact support and give a complete analysis of nonstationary tight wavelets frames. In this paper,we investigate some properties of pseudo splines as following,
·Give a regularity analysis of pseudo splines;
·Construct symmetric complex pseudo splines and symmetric or antisymmetric complex tight wavelets frames with high approximation order in L_2(R);
·Combine the definitions of B-splines and Box splines,we define pseudo box splines.Pseudo box splines are one multivariate generalization of univariate pseudo splines.
·With the masks of pseudo splines,we construct nonstationary symmetric C~∞complex pseudo splines and pseudo box splines.
Moreover,we investigate the convergence of cascade algorithm associated with the masks h(o|¨)lder continuous.We construct Riesz wavelet bases in L_2(R) with fast decay.We also investigate the support of a refinable vector satisfying an inhomogeneous refinement equation.By using some methods introduced by So and Wang, an estimate is given for the support of each component function of a compactly supported refinable vector satisfying an inhomogeneous matrix refinement equation with finitely supported masks.
引文
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[30]B.Han,Refinable functions and cascade algorithms in weighted spaces with H(o|¨)lder continuous masks,SIAM J.Math.Anal.40,(2008),70-102.
[31]B.Han,Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules,Adv.Comput.Math.(2008),doi:10.1007/s10444-008-9102-7.
[32]B.Han,Vector cascade algorithms and refinable function vectors in Sobolev spaces,J.Approx.Theory 124,(2003),44-88.
[33]B.Han and J.Hui,Compactly supported orthonormal complex wavelets with dilation 4 and symmetry Appl.Comput.Harmon.Anal.(2008),doi:10.1016/j.acha.2008.10.005.
[34]B.Han and Q.Mo,Multiwavelet frames from refinable function vectors Adv.Comput.Math.18,(2003),211-245.
[35]B.Han and Q.Mo,Symmetric MRA tight wavelet frames with three generators and high vanishing moments,Appl.Comput.Harmon.Anal.18,(2005),67-93.
[36]B.Han and Q.Mo,Tight wavelet frames generated by three symmetric B-spline functions with high vanishing moments,Proc.Amer.Math.Soc.132,(2004),77-86.
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[38]B.Han and Z.W.Shen,Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames,Israel J.Math.,to appear.
[39]B.Han and Z.W.Shen,Compactly supported symmetric C~∞ wavelets with spectral approximation order,SIAM J.Math.Anal.40,(2008),905-938.
[40]B.Han and Z.W.Shen,Wavelets with short support,SIAM J.Math.Anal.38,(2007),530-556.
[41]B.Han and Z.W.Shen,Wavelets from the Loop scheme,J.Fourier Anal.Appl.11,(2005),615-637.
[42]C.Heil and D.Colella,Matrix refinement equations:Existence and uniqueness,J.Fourier Anal.Appl.2,(1996),363-377.
[43]T.A.Hogan,Stability and indepeddence for multivariate refinable distribution,J.Approx.Theory 98,(1999),363-377.
[44]K.Hormann and M.A.Sabin,A family of subdivision schemes with cubic precision,Comput.Aided Geom.Design 25,(2008),41-52.
[45]R.Horn and C.Johson,Matrix Analysis,Cambirdge University Press,London,1989.
[46]K.Jetter and D.X.Zhou,Approximation order of linear operators and finitely generated shift-invariant spaces,(1998),(preprint).
[47]R.Q.Jia,Cascade algorithms in wavelet analysis,in:D.X.Zhou(Ed.),Wavelet Analysis:Twenty Years' Developments,World Scientific,Singapore,(2002),196-230.
[48]R.Q.Jia,Subdivision schemes in L_p spaces,Adv.Comput.Math.3,(1995),309-341.
[49]R.Q.Jia,Q.T.Jiang and S.L.Lee,Convergence of cascade algorithm in Sobolev spaces and integrals of wavelets,Numer.Math.91,(2002),453-473.
[50]R.Q.Jia,Q.T.Jiang,and Z.W.Shen,Convergence of cascade algorithms associated with nonhomogeneous refinement equations,Proc.Amer.Math.Soc.129,(2001),415-427.
[51]R.Q.Jia,Q.T.Jiang,and Z.W.Shen,Distributional solutions of nonhomogeneous discrete and continuous refinement equations,SIAM J.Math.Anal.32,(2000),420-434.
[52]R.Q.Jia and C.A.Micchelli,Using the refinement equations for the construction of pre-wavelets,Ⅱ.Powers of two,Curves and Surfaces,Academic Press,Boston,MA.(1991),209-246.
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