广义多元超复函数不等式及其应用
Generalized Multivariate Hypercomplex Function Inequalities and Their Applications
AI总结：
本文将 Mond-Pecaric 方法推广到多算子自变量的多元超复函数，利用基于 Cybenko 逼近理论的 sigmoid 函数构造任意误差阈值的上下界，导出比率型与差型新不等式，并提出 W-有界性概念及随机张量界代数。
摘要：
本工作将 Mond-Pecaric 方法推广到以多个算子为自变量的函数，方法是对原函数提供任意接近的逼近。不同于以往研究使用线性函数建立多元函数的下界和上界，我们基于 Cybenko 提出的多元函数逼近方法，应用 sigmoid 函数来实现具有任意指定误差阈值的这些界。该方法使我们能够导出多元超复函数的基本不等式，从而得到基于比率型和差型的新不等式。对于这些新导出的多元超复函数不等式的应用，我们首先通过应用比率型多元超复不等式，引入了一个称为超复函数 W-boundedness（W-有界性）的新概念。W-有界性推广了输入来自 Banach 空间的范数映射的 R-boundedness（R-有界性）。此外，我们发展了多元超复函数的逼近理论，并建立了界代数，包括多元随机张量的算子界和尾界代数。
英文摘要：
This work extends the Mond-Pecaric method to functions with multiple operators as arguments by providing arbitrarily close approximations of the original functions. Instead of using linear functions to establish lower and upper bounds for multivariate functions as in prior work, we apply sigmoid functions to achieve these bounds with any specified error threshold based on the multivariate function approximation method proposed by Cybenko. This approach allows us to derive fundamental inequalities for multivariate hypercomplex functions, leading to new inequalities based on ratio and difference kinds. For applications about these new derived inequalities for multivariate hypercomplex functions, we first introduce a new concept called W-boundedness for hypercomplex functions by applying ratio kind multivariate hypercomplex inequalities. W-boundedness generalizes R-boundedness for norm mappings with input from Banach space. Additionally, we develop an approximation theory for multivariate hypercomplex functions and establish bounds algebra, including operator bounds and tail bounds algebra for multivariate random tensors.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
