A piecewise Pad\'e-Chebyshev type (PiPCT) approximation method is proposed to minimize the Gibbs phenomenon in approximating piecewise smooth functions. A theorem on $L^1$-error estimate is proved for sufficiently smooth functions using a decay property of the Chebyshev coefficients. Numerical experiments are performed to show that the PiPCT method accurately captures isolated singularities of a function without using the positions and the types of singularities. Further, an adaptive partition approach to the PiPCT method is developed (referred to as the APiPCT method) to achieve the required accuracy with a lesser computational cost. Numerical experiments are performed to show some advantages of using the PiPCT and APiPCT methods compared to some well-known methods in the literature.

翻译：提议了一种Papewith Pad\'e-Chebyshev 类型(PiPCT)近似法,以在近似平滑功能中最大限度地减少Gibs现象。使用Chebyshev 系数的衰变属性,证明以$L$1$-error 估计的理论可以充分顺利地发挥作用。进行了数值实验,以表明PiPCT 方法精确地捕捉了一个函数的孤立特性,而没有使用位置和独特性的类型。此外,还制定了PiPCT 方法的适应性分割法(称为APiPCT 方法),以便以较低的计算成本达到所要求的准确性。进行了数值实验,以表明使用PiPCT和APiPCT方法与文献中一些广为人知的方法相比具有一些优势。