Approximation theorem is one of the most important aspects of numerical analysis that has evolved over the years with many different approaches. Some of the most popular approximation methods include the Lebesgue approximation theorem, the Weierstrass approximation, and the Fourier approximation theorem. The limitations associated with various approximation methods are too crucial to ignore, and thus, the nature of a specific dataset may require using a specific approximation method for such estimates. In this report, we shall delve into Chebyshev's polynomials interpolation in detail as an alternative approach to reconstructing signals and compare the reconstruction to that of the Fourier polynomials. We will also explore the advantages and limitations of the Chebyshev polynomials and discuss in detail their mathematical formulation and equivalence to the cosine function over a given interval [a, b].

翻译：逼近理论是数值分析中最核心的课题之一，多年来已发展出多种不同方法。最常用的逼近方法包括勒贝格逼近定理、魏尔斯特拉斯逼近定理和傅里叶逼近定理。不同逼近方法的局限性不容忽视，特定数据集的性质可能需要采用特定的逼近方法进行估计。本报告将详细探讨切比雪夫多项式插值作为信号重建的替代方法，并将其重建效果与傅里叶多项式进行对比。我们还将分析切比雪夫多项式的优势与局限性，详细讨论其数学推导过程，以及在给定区间[a, b]上该多项式与余弦函数的等价性。