Bernstein Polynomial Approximation
Approximate continuous functions with nonnegative Bernstein weights and compare the approximation error as degree grows.
Free theory articleApproximate continuous functions with nonnegative Bernstein weights and compare the approximation error as degree grows.
Concept & interpretation
The Bernstein polynomial approximation provides a constructive way to prove the Weierstrass Approximation Theorem. For any continuous function f on the interval [0, 1], we can construct a sequence of polynomials that converges uniformly to f. The n-th Bernstein polynomial for f is defined as:
Experimenting with the Approximation
You can investigate how the degree and target function affect the approximation in the interactive module: Bernstein Polynomial Approximation.
Try the following steps to observe the convergence properties:
- Select the V-shape target function, which has a sharp cusp at x = 0.5.
- Set the Degree n to a low value (like 2) and observe the large absolute error at the Probe x of 0.5.
- Gradually increase the Degree n up to 12. You will see the red approximation curve become tighter around the cusp, reducing the absolute error, even though Bernstein polynomials converge slowly for non-smooth functions.
- Check the boundary points by setting Probe x to 0 or 1. Notice that the approximation is exact at the endpoints because the basis weights cleanly resolve to 1 and 0.