arXiv math/06063192006-06-13math.FA

符合规定规范的紧框架完成

Tight frame completions with prescribed norms

P. Massey, M. Ruiz

摘要：

设$\hil$是一个有限维（实或复）Hilbert空间，$\{a_i\}_{i=1}^\infty$是一个非增的正数序列.给定$\hil$中的有限向量序列$\f$，我们找到了$\hil$中存在$r\in \NN\cup\{\infty\}$和贝塞尔序列$\g$的充要条件，使得$\cF\cup\cG$是$\hil$的紧框架，$\|g_i\|^2=a_i$是$1\leq i\leq r$的紧框架.此外，在这种情况下，我们使用此属性计算最小值$r\in \NN\cup\{\infty\}$。利用最近的Schur-Horn定理的结果，我们也得到了一个不是最优的，但算法可计算（在有限的步骤数）紧完成序列$\cG$。

英文摘要：

Let $\hil$ be a finite dimensional (real or complex) Hilbert space and let $\{a_i\}_{i=1}^\infty$ be a non-increasing sequence of positive numbers. Given a finite sequence of vectors $\f$ in $\hil$ we find necessary and sufficient conditions for the existence of $r\in \NN\cup\{\infty\}$ and a Bessel sequence $\g$ in $\hil$ such that $\cF\cup\cG$ is a tight frame for $\hil$ and $\|g_i\|^2=a_i$ for $1\leq i\leq r$. Moreover, in this case we compute the minimum $r\in \NN\cup\{\infty\}$ with this property. Using recent results on the Schur-Horn theorem, we also obtain a not so optimal but algorithmic computable (in a finite numbers of steps) tight completion sequence $\cG$.

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