Borel extension of a prescribed smooth jet

Statement

Let $U\subseteq\mathbb R^n$ be open and $a_j\in C^\infty(U)$ for every integer $j\ge0$ . There exists $F\in C^\infty(U\times\mathbb R)$ such that

\partial_t^jF(x,0)=a_j(x)\qquad(j\ge0).

This is Borel's smooth jet extension theorem, with $x$ as a parameter. No growth bound on the sequence $(a_j)$ is required.

Shrinking-cutoff construction

Choose $\chi\in C_c^\infty(\mathbb R)$ , supported in $[-1,1]$ , equal to one near zero. Set

F(x,t)=\sum_{j=0}^\infty \chi(t/\varepsilon_j)\,a_j(x)\frac{t^j}{j!}.

Take a compact exhaustion $(K_j)$ of $U$ . For each $j\ge1$ , choose $0<\varepsilon_j\le2^{-j}$ so that every mixed derivative of total order at most $\lfloor j/2\rfloor$ of its summand has supremum at most $2^{-j}$ on $K_j\times\mathbb R$ . This is possible: for a fixed number $q<j$ of normal derivatives, the cutoff scaling and product rule bound the summand by a constant times $\varepsilon_j^{j-q}$ .

For each fixed compact set and derivative order, all sufficiently late terms obey the geometric bound. The smooth-series criterion proves convergence with every derivative. At $t=0$ , the cutoff is constant near zero, and only the $j$ -th polynomial contributes to the $j$ -th normal derivative.

Support and nonuniqueness

Take $\varepsilon_0\le1$ ; then the construction is supported in $|t|\le1$ . If all $a_j$ are supported in one compact $K\subset U$ , so is the sum in the spatial variable. Two realizations of the same jet differ by a function flat on $t=0$ . A formal Taylor series need not converge even though a smooth realization always exists.

References

Richard Melrose, Lectures on Pseudodifferential Operators, Lecture 2 (Borel summation).

General increasing-order series

Asymptotic summation by shrinking cutoffs allows increasing real decay orders, logarithmic factors, and fixed derivative losses. Extension from bounded endpoint jets applies this theorem to a function already defined on one side of an endpoint.