论文标题
关于可分离的希尔伯特空间的张量产物中序列的最小总和
On the minimal Sums of sequences in the tensor product of separable Hilbert spaces
论文作者
论文摘要
众所周知，在两个可分离的希尔伯特空间的张量产物中，两个序列的张量产物是框架，并且仅当该产品的每个组件都是框架时。本文提出了一种对上述结果的概括，通过处理有限数量序列的张量产物的有限总和的序列S。当且仅当它是每个项是贝塞尔序列的张量产物时，我们才证明s是贝塞尔序列。我们还规定了S成为框架的必要条件。对于高于一个的尺寸，我们推断出有限秩方函数生成的Gabor系统的几个结果。同时，其中一些结果的一维版本令人惊讶地证明或不赞成。
It is known that the tensor product of two sequences, in the tensor product of two separable Hilbert spaces, is a frame if and only if each component of that product is a frame. This paper proposes a sort of generalization of the aforementioned result by dealing with sequences S that are finite minimal sums of tensor products of a finite number of sequences. We prove that S is a Bessel sequence if and only if it is a sum for which each term is the tensor product of Bessel sequences. We also state necessary conditions for S to be a frame. For dimensions higher than one, we deduce several results on Gabor systems generated by finite rank square integrable functions. Meanwhile, the one dimensional versions of some of these results are surprisingly extremely difficult to prove or disapprove.
