算子值域空间下信号完全恢复的矩阵方法
Matrix Methods for Perfect Signal Recovery Underlying Range Space of Operators
摘要：
本文的主要目的是用矩阵方法研究有限维希尔伯特空间中算子的完备重构。为此，我们首先得到了标准K-对偶的更多结构。%，并考察了概率擦除下的最优K-对偶问题。然后，我们研究了当删除集满足最小冗余条件或K-框架具有最大鲁棒性时信号的恢复和鲁棒性问题。此外，我们表明，差错率降低擦除下，如果$K$ -帧是均匀过剩。针对编码帧（K-dual）的防擦除保护问题，引入了$(r,k)$ -矩阵的概念来恢复丢失的数据，并通过矩阵方程解决了完全恢复问题。利用算子译码框架上的最小冗余条件，讨论了这类矩阵的存在性。最后，我们展示了几个例子，说明我们的结果和使用新的矩阵相对于以前的方法在存在和建设的优势。
英文摘要：
The most important purpose of this article is to investigate perfect reconstruction underlying range space of operators in finite dimensional Hilbert spaces by matrix methods. To this end, first we obtain more structures of the canonical K-dual. % and survey optimal K-dual problem under probabilistic erasures. Then, we survey the problem of recovering and robustness of signals when the erasure set satisfies the minimal redundancy condition or the K-frame is maximal robust. Furthermore, we show that the error rate is reduced under erasures if the $K$-frame is of uniform excess. Toward the protection of encoding frame (K-dual) against erasures, we introduce a new concept so called $(r,k)$-matrix to recover lost data and solve the perfect recovery problem via matrix equations. Moreover, we discuss the existence of such matrices by using minimal redundancy condition on decoding frames for operators. Finally, we exhibit several examples that illustrate our results and the advantage of using the new matrix with respect to previous approaches in existence and construction.
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