ON THE WHITNEYS EXTENSION THEOREM

We prove an extension theorem which completes the WHITNEY'S extension theorem and we generalize the ASCOLI'S theorem to Taylorian fields. We formulate the notion of stability of a HERMITE extension scheme and p rove that a stable HERMITE extrapolation scheme (the domains of the Ta ylorian fields to extend are discrete and closed) needs to be the rest riction of a stable HERMITE extension scheme. We prove that the origin al scheme yielded by the WHITNEY'S proof is unstable and we modify it to improve its stability.