Lemma 11.29.686B1F5EAC034

Proof. The normalized rank-one projector $Y$ determines the point $[t]$ of projective space, and its differential vanishes only in directions in which $\mathrm dt$ is proportional to $t$. Since $A$ is very ample, $Y$ is therefore injective with injective differential, and compactness makes it an embedding. A $C^2$ function on this image extends to a $C^2$ function on an ambient cube: use local coordinates in which the image is a graph, extend constantly in the normal coordinates, and combine these extensions with a finite smooth partition of unity and a cutoff. Rescale the cube to $[0,1]^ q$.

For such an extension $f$, its multivariate Bernstein polynomial is

where $j=(j_1,\ldots ,j_ q)$. For a multi-index $\alpha $ with $|\alpha |\leq 2$, differentiation expresses $\partial ^\alpha B_ Nf$ as an average of $\Delta _{1/N}^\alpha f(j/N)$, with prefactor $\prod _\ell (N)_{\alpha _\ell }$ and Bernstein degrees $N-\alpha _\ell $. Here $(N)_ r=N(N-1)\cdots (N-r+1)$, and $\Delta ^\alpha _{1/N}$ is the corresponding iterated forward difference with step $1/N$. Repeated use of the fundamental theorem of calculus expresses $N^{|\alpha |}\Delta _{1/N}^\alpha f(j/N)$ as an average of $\partial ^\alpha f$ at points within $2/N$ in each affected coordinate.

The remaining Bernstein weights are the law of independent binomial variables of degrees $N-\alpha _\ell $, divided by $N$. Their means differ from $y_\ell $ by at most $2/N$ and their variances are at most $1/(4N)$. Uniform continuity of $\partial ^\alpha f$, splitting the average into a fixed small neighborhood of $y$ and its complement, and Chebyshev's inequality give uniform convergence to $\partial ^\alpha f$. The prefactor divided by $N^{|\alpha |}$ tends to one. This proves convergence for every derivative of order at most two, including the function itself. Restrict to the embedded image and use the chain rule in a finite collection of charts to obtain the asserted $C^2$ approximation. $\square$