In Example 1, we describe a subset X of the plane and a function on X which has a ${}^k$-extension to the whole ${ℝ}^2$ for each finite, but has no ${}^{\infty }$-extension to ${ℝ}^2$. In Example 2, we construct a similar example of a subanalytic subset of ${ℝ}^5$; much more sophisticated than the first one. The dimensions given here are smallest possible.

We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces ${}_{\left(\right)}\left({[-1,1]}^r\right)$; (b) there is no continuous linear extension map from ${\Lambda }_{\left(\right)}^{\left(r\right)}$ into ${}_{\left(\right)}\left({ℝ}^r\right)$; (c) under some additional assumption on , there is an explicit extension map from ${}_{\left(\right)}\left({[-1,1]}^r\right)$ into ${}_{\left(\right)}\left({[-2,2]}^r\right)$ by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

For a large class of Hardy fields their extensions containing non-${}^{\infty }$-germs are constructed. Hardy fields composed of only non-${}^{\infty }$-germs, apart from constants, are also considered.

We extend a result of M. Tamm as follows:Let $f:A\to ℝ,A\subseteq {ℝ}^{m+n}$, be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x\mapsto x^r:(0,\infty )\to ℝ,r\in ℝ$. Then there exists $N\in \mathbb{N}$ such that for all $(a,b)\in A$, if $y\mapsto f(a,y)$ is $C^N$ in a neighborhood of $b$, then $y\mapsto f(a,y)$ is real analytic in a neighborhood of $b$.

In [3], J. Chaumat and A.-M. Chollet prove, among other things, a Whitney extension theorem, for jets on a compact subset E of ℝⁿ, in the case of intersections of non-quasi-analytic classes with moderate growth and a Łojasiewicz theorem in the regular situation. These intersections are included in the intersection of Gevrey classes. Here we prove an extension theorem in the case of more general intersections such that every $C^{\infty }$-Whitney jet belongs to one of them. We also prove a linear extension theorem...