Abstract

The atomic decomposition of signals is one of the most important problems in the frame theory. K-dual frame pairs may be used to stably reconstruct elements from the range of bounded linear operators on Hilbert spaces. The purpose of this paper is making K-dual frame pairs and finding common K-dual Bessel sequence. We present a sufficient condition on operators on $\mathcal{H}$ which takes a K-dual frame pairs to other ones; characterize bounded linear operators on $l^2(\mathcal{J})$ that transform K-dual frame pairs to other ones; prove that two Bessel sequences can always be extended to a K-dual frame pair, and that two orthogonal K-frames have a common K-dual Bessel sequence under certain conditions; and obtain a sufficient condition which the K-duals of one K-frame is contained in the ones of another K-frames. Abundant examples are also provided to illustrate the generality of the theory.

摘要

信号的原子分解是框架理论中最重要的问题之一。K对偶帧对可用于从希尔伯特空间上的有界线性算子的范围稳定地重构元素。本文的目的是制作K对偶帧对，并找到共同的K对偶贝塞尔序列。我们为运营商提供了充分的条件$\mathcal{H}$这需要将K对偶帧对与其他对偶帧对进行配对；刻画有界线性算子${升}^{\mathrm{2个}}（\mathcal{Ĵ}）$将K对偶帧对转换为其他对；证明两个贝塞尔序列始终可以扩展为一个K对偶帧对，并且在某些条件下，两个正交K帧具有相同的K对偶贝塞尔序列。并获得一个充分的条件，即一个K帧的K对偶包含在另一个K帧的K对偶中。还提供了大量示例来说明该理论的普遍性。