We extend the idea of approximating piecewise smooth univariate functions using rational approximation introduced in \cite{aka_bas-19a} to two-dimensional space. This article aims to implement the novel piecewise Maehly based Pad\'e-Chebyshev approximation \cite{mae_60a}. We first develop a method referred to as PiPC to approximate univariate piecewise smooth functions and then extend the same to a two-dimensional space, leading to a bivariate piecewise Pad\'e-Chebyshev approximation (Pi2DPC) for approximating piecewise smooth functions in two-dimension. We study the utility of the proposed techniques in minimizing the Gibbs phenomenon while approximating piecewise smooth functions. The chief advantage of these methods lies in their non-dependence on any apriori knowledge of the locations and types of singularities (if any) present in the original function. Finally, we supplement our methods with numerical results to validate their effectiveness in diminishing the Gibbs phenomenon to negligible levels.

翻译：我们利用在\ cite{aka_bas-19a} 中引入的理性近似法,将近似平滑的单向单向功能的概念推广到二维空间。 文章的目的是执行小巧的Maehly Pad\' e- Chebyshev 近似法\ cite{mae_ 60a}。 我们首先开发了一种称为PiPC 的方法, 以近似单向单向光滑功能, 然后将同样的方法扩大到二维空间, 导致双向的pad\'e- Chebyshev 近似法( Pi2DPC) 近似双向平滑函数( Pi2DPC) 。 我们研究了拟议技术在尽量减少吉卜斯现象的同时, 与近似平滑函数的近似法的实用性。 这些方法的主要优势在于它们不依赖原始函数中存在的任何关于奇特地点和奇特类型( 如果有的话) 的任何原始知识。 最后, 我们用数字结果来补充我们的方法, 来验证它们在将吉布斯现象降到微不足道的水平上的有效性。