On the Computation of the General Simplicial Bernstein Inclusion–Isotone Property for Stability Analysis of Least-Squares Polynomials

Bernsteinpolynomials on simplices provide a powerful framework for approximation, geometric modeling, and numerical analysis. A fundamental structural feature of their coefficients is the inclusion–isotone property: under subdivision of the domain simplex, the Bernstein coefficients over subsimplices remain confined within the bounds of the original coefficients. While this behavior has been established previously only for particular cases, a general and unified treatment has been lacking. In this paper, we present a dimension-independent proof of inclusion isotonicity for general simplicial Bernstein polynomials based on the positivity and stochasticity of the subdivision operator induced by barycentric refinement. We further show that the resulting Bernstein bounds over subsimplices are sharp, in the sense that they accurately represent the attainable range of the polynomial on the refined domain. In addition, we develop an optimized least-squares approximation in the Bernstein–Bezier basis, which improves numerical stability compared to monomial representations and admits an efficient computational implementation. Finally, nonlinear examples and applications to Lyapunov-based stability analysis of polynomial systems illustrate the contraction of coefficient bounds under subdivision and demonstrate the practical effectiveness of the proposed approach.