Topological and geometric properties of refinable functions and MRA affine frames
Topological and geometric properties of refinable functions and MRA affine frames
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DOI：
10.1016/j.acha.2010.04.002
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发表时间：
2011-03
影响因子：
2.5
通讯作者：
D. Han;Qiyu Sun;Wai-Shing Tang
中科院分区：
文献类型：
--
作者：
D. Han;Qiyu Sun;Wai-Shing Tang
我们研究了L2（Rd）中所有可加细函数的集合R和所有MRA仿射框架的集合R的一些拓扑和几何性质。证明了R在L2（Rd）中无处稠密; R的单位球面在L2范数下是路径连通的;对任意由L2-函数f0，.，fM生成的M维超平面，要么几乎所有超平面上的函数都是可加细的，要么几乎所有超平面上的函数都是不可加细的.我们证明了所有MRA仿射框架的集合在L2（Rd）中无处稠密。我们还得到了R的L2-闭包R <$的一个新的刻画，并将上述拓扑和几何结果从R推广到R <$，甚至进一步推广到所有可加细向量集及其L2-闭包.
We investigate some topological and geometric properties of the set R of all refinable functions in L2(Rd), and of the set of all MRA affine frames. We prove that R is nowhere dense in L2(Rd); the unit sphere of R is path-connected in the L2-norm; and for any M-dimensional hyperplane generated by L2-functions f0,…,fM, either almost all the functions in the hyperplane are refinable or almost all the functions in the hyperplane are not refinable. We show that the set of all MRA affine frames is nowhere dense in L2(Rd). We also obtain a new characterization of the L2-closure R¯ of R, and extend the above topological and geometric results from R to R¯, and even further to the set of all refinable vectors and its L2-closure.