The Bernstein polynomial basis was developed by Bernstein to prove the Weierstrass theorem.
Convenience. Orthogonal polynomials tend to be better behaved, at least in L^2 norms. Chebyshev polynomials are better to use in the max or infinity norms. But we use the standard monomial basis for convenience of calculation. For low-degree problems the lesser stability of this basis is not a problem.
In the fields of Computer Aided Design, Computer Graphics, and Computer Aided Geometric Design, the Bernstein polynomial basis makes the basics of CAD systems.
The problem is that building a software on the monomial basis accumulates the error.
The orthogonal polynomials have special properties in finding the polynomials of best approximation in the infinity(T_n), first(U_n), and second norms(orthogonal polynomial). In particular, they simplify the computations and characterization.
Since Vandermonde matrix in most of the cases is ill-conditioned, then monomial basis almost always gives inaccurate results. On the other hand other polynomials are more complicated computationally.
De Casteljau Algorithm simplifies generating a curve using a cheap technique. The resulting curve is written in terms of Bernstein polynomial basis. This algorithm is used in many CAD systems. However, doing more research might simplify and make this technique more popular.
The Bernstein polynomials are adapted to convex linear combinations. For other uses other polynomials may be more suitable. There is a large number of classical families of orthogonal polynomials in [0,1], each adapted to a particular Sturm-Liouville problem. The powers of x are adapted to abstract algebra.
Each polynomial family has its uses and one should not expect any one polynomial family to be universally useful.
Davis Stewart has given a cogent answer to the question for this thread.
I thought perhaps the readers of this thread would find the attached paper by Rida Farouki, who provides a 100 year perspective on the Bernstein polynomial basis.
Thank you James for the source paper; it tells the story of Bernstein polynomials and their importance in computations.
In favour of the normal powers 1,z,z²,z³,... one should also take into account their role in the definition of holomorphic functions and in function theory in general.
@Ulrich Mutze: There has to be a place for the canonical basis of the free abelian algebra on one generator...
Dear all,
Thank you for sharing your thoughts; it is highly appreciated. The choice of a basis affects the results in many ways.
because the second one is more common and easier to understand and more interpretative.
I thank Prof.James F Peters for the reference on Bernstein polynomial basis.
-Sundar
Dear @Ahmad, thank you for sharing the forum; the monomial basis are convenient in calculations and we are used to use them, while the Bernstein polynomial basis have geometric interpretations.
Dear @Sundarapandian, thank you for sharing the forum; it is good to learn about the Bernstein polynomial basis because they are the basics for many softwares.
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