论文标题
Hilbert C* - 模型中的B-Spline插值问题
B-spline interpolation problem in Hilbert C*-modules
论文作者
论文摘要
我们介绍了$ b $ -SPLINE插值问题，该问题对应于Hilbert $ C^*$ - 模块上的$ C^*$ - 有价值的sesquilinear形式，并研究其基本属性以及解决方案的独特性。我们首先在Hilbert $ c^*$ - 模块是自动划分的情况下研究问题。将有限的$ c^*$ - 在希尔伯特$ c^*$ - 模块上将有价值的sesquilinear形式扩展到其第二个双重的sesquilinear形式，然后我们为$ b $ - $ - $ SPLINE插值问题提供了一些必要和足够的条件，以提供解决方案。传递到Hilbert $ w^*$ - 模块的设置，我们通过表征何时表征了何时何时表征扩展的$ c^*$ - 对双$ \ Mathscr {x}'Hilbert $ w^*$ W^*$ module $ \ Mathscr的$ \ Mathscr {x}'$的估价的sesquilinear。结果，我们给出了足够的条件，即对于一个自dual hilbert $ w^*$ - module $ \ mathscr {x} $的正交补充的子模块，相对于另一种$ c^*$ - $ \ nertsody在$ \ mathscr {x} $上是正交的。最后，广泛讨论了$ c^*$ - $ c^*$的模块的$ b $ -spline插值问题的解决方案。提供了几个示例来说明问题的存在或缺乏解决方案。
We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^*$-module is self-dual. Extending a bounded $C^*$-valued sesquilinear form on a Hilbert $C^*$-module to a sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the $B$-spline interpolation problem to have a solution. Passing to the setting of Hilbert $W^*$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^*$-valued sesquilinear to the dual $\mathscr{X}'$ of the Hilbert $W^*$-module $\mathscr{X}$ has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^*$-module $\mathscr{X}$ is orthogonally complemented with respect to another $C^*$-inner product on $\mathscr{X}$. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^*$-modules over $C^*$-ideals of $W^*$-algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem.
