Uniform Error Bounds for Bernstein Polynomial Approximation

Exact quadratic and cubic identities with tolerance-based order selection

Abstract

Bernstein operators provide a constructive approximation scheme whose order controls sampling density and approximation error. For quadratic and cubic monomials on the unit interval, we derive exact error polynomials from binomial factorial moments and determine their uniform maxima. The quadratic maximum occurs at the midpoint, whereas the cubic maximizer depends on the order and approaches two thirds. These identities yield explicit checks of the first-order asymptotic error and a direct procedure for selecting an operator order from a prescribed uniform tolerance. Numerical comparisons distinguish operator order from the algebraic degree of the resulting polynomial and compare analytic maxima with sampled curves. The cubic example shows why a midpoint evaluation does not determine a uniform error. Formulas, critical points, and tolerance thresholds are supplied together, providing a reproducible comparison of elementary polynomial targets within the classical theory of positive approximation operators.