We prove two theorems of N. Kuiper by standard methods of the calculus on Banach spaces. C0-sufficiency for Cr-functions of the r-jet of a real valued Cr-function of lowest degree r0 ≥ 2 on a Hilbert space follows from Kuiper's condition Q(r) also on a Banach space endowed with smooth partitions of unity. A new proof is given of the C1-sufficiency for Cr + p-functions, p ≥ r - r0, of the r + p-jet of a Q(r)-function on Hilbert space that also holds on Banach space modulo a further condition on the behavior of the function near its degenerate critical point.
ASJC Scopus subject areas
- Applied Mathematics