If I is an any open interval in R containing 0 (possibly I = R), then there exists a complex-valued smooth function F(t, x) defined on I×U, such that
for k ≥ 0 and x in U.
Note that it suffices to prove the result for a small interval I = (−ε,ε), since if ψ(t) is a smooth bump function with compact support in (−ε,ε) equal identically to 1 near 0, then ψ(t) ⋅ F(t, x) gives a solution on R × U. Similarly using a smooth partition of unity on Rn subordinate to a covering by open balls with centres at δ⋅Zn, it can be assumed that all the fm have compact support in some fixed closed ball C. For each m, let
where εm is chosen sufficiently small that
for |α| < m. These estimates imply that each sum
is uniformly convergent and hence that
is a smooth function with
Note: Exactly the same construction can be applied, without the auxiliary space U, to produce a smooth function on the interval I for which the derivatives at 0 form an arbitrary sequence.
- Erdélyi, A. (1956), Asymptotic expansions, Dover Publications, pp. 22–25, ISBN 0486603180
- Golubitsky, M.; Guillemin, V. (1974), Stable mappings and their singularities, Graduate texts in Mathematics 14, Springer-Verlag, ISBN 0-387-90072-1
- Hörmander, Lars (1990), The analysis of linear partial differential operators, I. Distribution theory and Fourier analysis (2nd ed.), Springer-Verlag, p. 16, ISBN 3-540-52343-X