STABILITY AND CONVERGENCE OF UNIVARIATE L AGRANGE EXTENSION SCHEMES BY C-M FUNCTIONS

We formulate notions of stability and of convergence for univariate La grange extension schemes by C-m -functions, m greater than or equal to 1. We show that the extension schemes of the literature are unstable and we prove, for m = 1, that stable and convergent schemes exist. We generalize the original Whitney's theorem to functions of unclosed dom ain, we establish an Ascoli's type theorem for such functions and show that any stable extrapolation (the domain of functions to extend is f inite) scheme must be the restriction of a stable extension scheme.