= Bernstein polynomial degree preservation
{c}
{title2=$\deg B_np=\deg p\quad(\deg p\leq n)$}
For $K$ with a binomial distribution of parameters $n,x$, the Bernstein image of a monomial is $B_n(x^m)=\mathbb E(K/n)^m$. Expand $K^m$ in the monic <falling factorial> <basis>. Since $\mathbb E K^{\underline r}=n^{\underline r}x^r$, the <leading coefficient> becomes $n^{\underline m}/n^m$, which is nonzero for $m\leq n$. Thus an actual degree-$m$ <polynomial> retains its degree, even though its lower <coefficients> generally change.
Back to article page
