Inequalities on the closeness of two vectors in a Parseval frame
Inequalities on the closeness of two vectors in a Parseval frame
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Parseval 框架中两个向量的接近度不等式
DOI：
10.1007/s13160-023-00578-7
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发表时间：
2023
影响因子：
0.9
通讯作者：
Morimoto Akira
中科院分区：
文献类型：
--
作者：
Ashino Ryuichi;Mandai Takeshi;Morimoto Akira
设H是Hilbert空间H中的一个Parseval框架，也就是说，每个都成立。已知的是，对于每个k，以及if\Documentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$$\Left\|f_{k_0}\right\|=1$$\end{Document}。因此，我们可能会认为，如果接近1，那么彼此之间的角度就是接近的，也就是说，彼此之间不是那么“接近”。我们想通过一些不等式来定量地阐明这一点。事实上，我们可以证明以下不等式：\DocentClass[12pt]{Minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$\Begin{Align}|\langf_k，F_L|\le\Sqrt{1-\Vert f_k\Vert^2}\cot\Sqrt{1-\Vert f_L\Vert^2}\qquad\Text{对于L，\end{aliged}$$\end{Document}，这意味着如果\Documentclass[12pt]{Minimal}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amsfonts}\usepackage{amsbsy}\usepackage{matrssy}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\{Begin Document}$\Left\|f_k\right\$\end{Document}为接近1，且如果\DocumentClass[12pt]{Minimal}\Usepackage{amsath}\usepackage{wa ysym}\usepackage{amsFonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{matrsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$$\Left\|f_L\Right\|$$\end{Document}不是那么小，那么两个国家就不那么“亲近”了。
Letbe a Parseval frame in a Hilbert spaceH, that is,holds for every. It is well known that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\| f_k\right\| \le 1$$\end{document} for everyk, and that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\| f_{k_0}\right\| =1$$\end{document} for some, thenfor every. Hence, we might expect that ifis near 1, then the angle betweenand anotheris near, that is,andare not so “close” to each other. We want to make it quantitatively clear by some inequalities. In fact, we can prove the following inequality: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\langle f_k, f_l \rangle | \le \sqrt{1-\Vert f_k\Vert ^2} \cdot \sqrt{1-\Vert f_l\Vert ^2} \qquad \text {for} k\ne l, \end{aligned}$$\end{document}which implies that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\| f_k\right\| $$\end{document} is near 1 and if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\| f_l\right\| $$\end{document} is not so small, thenandare not so “close” to each other.