从连分数到有理逼近的理论研究
From Continued Fractions to Rational Approximation: A Theoretical Study
摘要: 连分数理论作为数学分析中的重要工具，在数的表示、逼近理论及数值计算中发挥着独特作用。本文以连分数理论为基础，结合数值逼近中的有理逼近方法，系统梳理了从经典连分数理论到现代有理逼近的发展脉络。文章首先回顾连分数的基本性质及其在实数表示中的作用，继而探讨渐近分数作为最佳有理逼近的核心理论，最后引入Padé逼近等现代有理逼近方法，揭示连分数思想在分析逼近中的深刻影响。研究表明，连分数不仅提供了最佳有理逼近的理论基础，其构造思想和收敛性分析也为现代数值逼近方法的发展提供了重要启示。
Abstract: As an important tool in mathematical analysis, continued fraction theory plays a unique role in number representation, approximation theory, and numerical computation. This thesis, based on continued fraction theory and combined with rational approximation methods in numerical approximation, systematically reviews the development from classical continued fraction theory to modern rational approximation. The article first reviews the basic properties of continued fractions and their role in real number representation, then discusses the core theory of convergents as best rational approximations, and finally introduces modern rational approximation methods such as Padé approximation, revealing the profound influence of continued fraction ideas in analytic approximation. Research shows that continued fractions not only provide the theoretical foundation for best rational approximation, but their constructive ideas and convergence analysis also offer important insights for the development of modern numerical approximation methods.
文章引用：贺龙. 从连分数到有理逼近的理论研究[J]. 理论数学, 2026, 16(6): 72-81. https://doi.org/10.12677/pm.2026.166158

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