近似斜对偶框架
Approximate oblique dual frames
AI总结：
针对框架表示中斜对偶框架存在显式表达式难、性质不符需求及计算不精确等问题，本文提出近似斜对偶框架概念，研究其性质与刻画，给出平移不变子空间中存在条件，并通过B样条示例验证其更优的数值计算属性。
摘要：
在使用框架的表示中，当分析与合成必须在不同子空间中完成时，就会出现斜对偶性。在某些情况下，我们无法得到斜对偶的显式表达式；而在另一些情况下，仅存在一个不具备我们所需性质的斜对偶框架。此外，实际计算中也无法做到完全精确。为解决这些问题，本文首先在可分希尔伯特空间的框架下引入并研究了近似斜对偶框架的概念。我们给出了近似斜对偶框架的若干性质，并提供了不同的刻画方式。随后我们聚焦于L²(R)的平移不变子空间中的近似斜对偶框架，给出了关于生成元的不同条件以保证其存在性。我们通过一个由B样条生成的框架序列示例，说明了近似斜对偶框架在数值与计算层面的重要性，其中利用前述结果构造出的近似斜对偶框架比精确斜对偶框架具有更优的属性。我们还给出了近似误差的表达式并研究了其特性。
英文摘要：
In representations using frames, oblique duality appears in situations where the analysis and the synthesis has to be done in different subspaces. In some cases, we cannot obtain an explicit expression for the oblique duals and in others there exists only one oblique dual frame which has not the properties we need. Also, in practice the computations are not exact. To give a solution to these problems, in this work we introduce and investigate the notion of approximate oblique dual frames first in the setting of separable Hilbert spaces. We present several properties and provide different characterizations of approximate oblique dual frames. We focus then on approximate oblique dual frames in shift-invariant subspaces of L^2(R)and g ive different conditions on the generators that assure their existence. The importance of approximate oblique dual frames from a numerical and computational point of view is illustrated with an example of frame sequences generated by B-splines, where the previous results are used to construct approximate oblique dual frames which have better attributes than the exact ones. We provide an expression for the approximation error and study its behaviour.
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