The extension problem considered in this paper is of the type given below: Let \(K_1\) and \(K\) be compact convex sets such that \(\text{int} (K_1) \supset K\), and such that \(\text{int} (K)\neq \emptyset\) or \(K= \{0\}\) and let a sequence \((N_a)\) of positive numbers be given. Then characterize the sequences \((M_a)\) such that the following holds: There is \(C>0\) such that any function \(f\in C^\infty (K)\) with \(\text{Sup} \{|f^{(a)} (x)\mid:x\in K\}\leq C_1 N_{|a|}\) for any \(a\), can be extended to \(F\subset C^\infty (K_1)\) such that \[ \text{Sup} \{|F^{(a)} (x)|: x\in K_1\}\leq C_1 C^{|a|+1} M_{|a|}. \] Two different types of functions were considered. First, the author considered the Carleman- Komatsu type ultradifferentiable functions [\textit{H. Komatsu}, J. Fac. Sci., Univ. Tokyo Sect. IA 20, 25-105 (1973; Zbl 0258.46039), ibid. 24, 607-628 (1977; Zbl 0385.46027)]\ for the extension problem and then the case of the Beurling type ultradifferential functions [\textit{R. W. Braun}, \textit{R. Meise} and \textit{B. A. Taylor}, Result. Math. 17, No. 3/4, 206-237 (1990; Zbl 0735.46022)]\ was considered by sketching the modifications at the end of the paper. Also while solving this kind of problem, the local and punctual image [\textit{L. Ehrenpris}, `Fourier analysis in several complex variables' (1970; Zbl 0195.10401)]\ of the restriction mapping on several type of ultradifferentiable functions were determined. Section 1 (entitled Fourier transformation) deals with a precise Paley- Wiener theorem for ultradistributions defined on compact convex sets and classes of ultradifferentiable functions which are introduced by imposing bounds on the derivatives of the function. Section 2 deals with the extension of differential functions.