Advantage of Bernstein polynomial basis

Solution 1:

I assume your question is about the advantage of Bernstein polynomials compared with Legendre polynomials.

For numerical applications, there is no advantage. The convergence speed of Bernstein polynomials for continuous function approximation is very bad!

Usually, Bernstein polynomials are introduced in approximation theory lectures to prove the theorem of Weierstraß:

For each $f \in C[a, b]$ and each $\varepsilon > 0$ there is a polynomial $p$ so that $\|f − p\|_{\infty} < \varepsilon$.

The proof uses the fact that the Bernstein polynomials form a Korovkin sequence.

For practical applications, the Chebyshev polynomials are much more important than the Bernstein or Legendre polynomials.