Bernstein's Polynomials for f(x)=x and f(x)=x^2: Sequence and Formula

[Artusartos]

Find the sequence (B_nf) of Bernstein's polynomials in

a) f(x)=x and

b) f(x)=x^2

Answers (from my textbook):

a) B_nf(x) = x for all n.

b) B_nf(x) = x^2 + \frac{1}{n} x (1-x)

I know that the bernstein's polynomial is:

B_nf(x) = \sum_{k=0}^n f (\frac{k}{n}) \binom{n}{k} x^k (1-x)^{n-k}

...but I don't know how they got the answer from this...

 

[HallsofIvy]

Have you used that formula to calculate, say, B₀ through B₅ for f(x)= x and f(x)= x²? That should give you an idea.

 

[Artusartos]

Have you used that formula to calculate, say, B₀ through B₅ for f(x)= x and f(x)= x²? That should give you an idea.

But how can I calculate B_0? If I say n=0, then

B_nf(x) = \sum_{k=0}^n f (\frac{k}{0}) \binom{0}{k} x^k (1-x)^{0-k}

So f(k/0) is undefined?

 

[HallsofIvy]

Sorry. Clearly "B₀" is not defined so calculate B₁, B₂, etc.

For example, with f(x)= x,
B_1(x)= f(0)\begin{pmatrix}1 \\ 0\end{pmatrix}x^0(1- x)^1+ f(1)\begin{pmatrix}1 \\ 1\end{pmatrix}x^1(1- x)^0= 0(1- x)+ 1x= x