Research on error control method for polynomial approximation based on equal amplitude oscillation theorem

To address the issue that polynomial approximation methods strongly depend on the analytical form of the objective function, this study proposes a new minimax polynomial approximation method based on the Chebyshev equioscillation theorem. It constructs near-optimal solutions through discrete sample points, introduces a threshold relaxation strategy to locate equioscillation points, and establishes a coefficient solution framework based on linear equations. Experiments cover both noiseless and noisy data scenarios. Compared with methods such as the Least Squares Method (LSM), the results show that the maximum absolute error of the proposed method is effectively reduced, and it performs excellently under different data densities and distributions. This can provide theoretical support for extreme deviation suppression in engineering fields, especially in safety-critical systems.