非線形問題とヘルダーの不等式  [in Japanese] Nonlinear Problems and Hölder's Inequality  [in Japanese]

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Author(s)

    • 田中 久陽 TANAKA Hisa-Aki
    • 電気通信大学大学院情報理工学研究科 Graduate School of Informatics and Engineering, University of Electro-Communications

Abstract

ヘルダーの不等式は,1888年数学者のロジャーズと1889年ヘルダーにより独立にその基礎が見いだされ,以降,関数解析等の解析学の基本的不等式として日常的に多用されている.しかし,意外なことに,この不等式の物理的解釈の例は,2014 年にようやく知られるようになった.本稿は,この不等式が最近の非線形問題の未解決な問題のエレガントな解答を与え,更に情報通信分野にも浸透しつつあるTsallis(ツァリス)統計の一つの基礎にもなり,更に現代制御理論を補強する可能性について解説する.

Since the basis of Hölder's inequality was found by mathematicians, i.e., independently by Rogers in 1888 and Hölder in 1889, the inequality has been frequently utilized as a basic inequality in mathematical analysis such as functional analysis. However, surprisingly, no physical interpretation of the inequality was introduced until 2014. In this article, I show that the inequality gives an elegant solution to a recent open nonlinear problem. It is also asserted that the inequality can be a fundamental basis in Tsallis statistics, studied recently in information and communications technology. Furthermore, it is argued that the inequality can make a useful addition to modern control theory.

Journal

  • IEICE ESS Fundamentals Review

    IEICE ESS Fundamentals Review 9(3), 219-228, 2016

    The Institute of Electronics, Information and Communication Engineers

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