Chebyshev's Equioscillation Theorem

Chebyshev’s Equioscillation Theorem states that the best approximation of a continuous function by a polynomial of a certain degree is characterized by the approximation oscillating equally above and below the true function at specific points. These points, called extremal points, are where the error (difference between the approximation and the actual function) reaches its maximum magnitude. The polynomial "touches" the maximum error at these points, oscillating between positive and negative peaks. This balance ensures the approximation minimizes the worst-case deviation across the entire interval, making it the optimal uniform approximation in the sense of minimizing the maximum error.