11.29.5 Approximation with two derivatives84DD621CC41A

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. $\square$

Proof. For nesting, multiply the sections in $V_ m$ by each $t_\alpha $ and express the products in the basis of $V_{m+1}$. The identity $\sum _\alpha \mathrm e^{-\phi }|t_\alpha |^2=1$ writes $G_{m,B}$ as the sum of the corresponding weighted quadratic functions at level $m+1$, which combine into one Hermitian coefficient matrix at that level. Surjectivity of the multiplication map is not needed.

For density, write $p_{m_0}(v)=P_0v$ in a local frame of $F$, and put $L=\mathrm e^{-m_0\phi /2}P_0$. Global generation of $E\otimes A^{m_0}$ means that $L$ is injective, so $M=L^*L$ is positive definite. If a $C^2$ Hermitian form has matrix $H$, define

Then $L^*CL=H$. A change of frame of $F$ cancels in this expression; a change of frame of $A$ multiplies $L$ by a scalar of modulus one, which also cancels. Thus $C$ is a globally defined $C^2$ Hermitian matrix function on $X$.

Applying lemma 11.29.6 to its real coordinates in the vector space of Hermitian matrices, we obtain Hermitian matrix polynomials in the entries of $Y$ that converge to $C$ in $C^2$; their degrees may increase along the sequence. Let $C_ q(Y)$ be such a polynomial, of degree at most $q$. Every monomial of degree $k$ in $Y$ has the form $t^ I\bar t^ J/\rho ^ k$, with $|I|=|J|=k$. For a polynomial of degree at most $q$, multiply each numerator by $\rho ^{q-k}$ to use the common denominator $\rho ^ q$. Consequently

is a quadratic expression in the product sections $\sigma _ i t^ I$ of $E\otimes A^{m_0+q}$, with weight $\mathrm e^{-(m_0+q)\phi }$. Express these products in the basis of $V_{m_0+q}$ and replace the coefficient matrix by its Hermitian part, which does not change the real function. The result is an element of $\mathcal H_{m_0+q}$. Since $L^*CL=H$ and multiplication by the fixed smooth matrix $L$ is continuous in $C^2$, these elements converge to $H$. $\square$