Generalized Multivariate Hypercomplex Function Inequalities and Their Applications

Shih-Yu Chang
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Abstract

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.
广义多元超复变函数不等式及其应用
这项工作通过提供原始函数的任意近似值,将蒙德-佩卡里克方法扩展到了以多个运算符为参数的函数。我们不再像以前的工作那样使用线性函数来建立多元函数的上下限,而是根据 Cybenko 提出的多元函数逼近方法,使用 sigmoid 函数来实现这些具有任意指定误差阈值的上下限。通过这种方法,我们可以推导出多元超复杂函数的基本不等式,从而得出基于比率和差分类型的新不等式。关于这些新导出不等式在多元超复变函数中的应用,我们首先通过应用比类多元超复变不等式,引入了一个新概念,即超复变函数的 W 有界性。W 有界性概括了从巴拿赫空间输入的规范映射的 R 有界性。此外,我们还发展了多元超复函数的近似理论,并建立了边界代数,包括多元随机张量形式的算子边界和尾边界代数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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