I recently encountered Bernstein polynomials in an LA class. My text giving very little information, I looked on the web for more. I found this thirteen page PDF, which is the most informative text I can find that is freely available for download no cost.
I worked through it, but I can't follow Dr. Joy's argument on p. 9, where he tries to "show that each power basis element can be written as a linear combination of Bernstein polynomials," using "the degree elevation formulas and induction." I don't see how he got to that second step, or exactly how he's using "the induction hypothesis." He must've left out some details, maybe if they were included I could follow this.
Can anybody explain to me what he's doing?
A couple of my random thoughts.
It seems to me he could've substantiated the claim using an informal argument something like this: every power basis element, such as t^3, is a Bernstein polynomial in the Bernstein basis of its own degree. For instance, t^3 = B(3,3). The problem arises when we're using a Bernstein basis of a higher degree, like B4 or B5. But the power-raising formulas guarantee that we can represent t^3 as a linear combination of B4 or B5 or any other Bernstein basis elements. The induction, I gather, comes in because strictly speaking we have to raise the powers one at a time. That seems to be his general idea, but he's trying to do it in a more techie and more exact way.
Another thing. One can make a matrix that will convert polynomials in the power basis (of a given degree) to the Bernstein basis (of the same degree). Such a matrix is easy to find, and this pdf even shows two examples. Such matrices display a simple pattern, which makes it easy to construct them as needed. These matrices are square and invertible, and the inverses will convert from the power basis to the Bernstein basis (again, same degree). With a matrix like that, it should be easy to convert the elements of the power basis, if anyone needs to do that.