Polynomial Approximation: Stability and Error Analysis

This lecture covers practical considerations for calculating the best polynomial approximation in the least squares sense, discussing the challenges of polynomial approximation, stability issues near interval boundaries, and the impact of noise on measurements. Different strategies to enhance the stability of polynomial interpolation are explored, such as using Chebyshev or Crencher-Cortis points, piecewise interpolation, and low-degree polynomial approximations. The lecture also delves into error analysis, discussing the precision of finite difference formulas for numerical differentiation and the convergence properties of different finite difference schemes.