Whitney’s extension theorem for ultradifferentiable functions of Beurling type

Abstract

d~ is defined similarly (replacing the all quantifier over h by an existence quantifier). Continuing the classical work of E. Borel [5] many authors (see Bronshtein [7], Bruna [8], Carleson [9], Dzanasija [10], Ehrenpreis [11], Komatsu [15], Mityagin [22], Petzsche [24] and Wahde [30]) have investigated conditions on (Mp)g~N ~ and on sequences (a~)~N~ implying the existence of fCg~u~)(R (resp. N{M~} (R) with f ~ ' ) ( 0 ) = a ~ for all ~CN~+ In the present note we study this question and a version of Whitney's extension theorem for the non-quasianalytic classes doo,(R which have been introduced by Beurling [2] and Bj/Srck [3] using the Fourier transform. Most familiar function classes, like the Gevrey classes, can be obtained by both methods (Mp=(pI) s or o9(x)= ]x] l/s, s > 1). However, in general, the two definitions lead to different classes. To define d%(R) we vary Beurling's approach a bit. We assume that o9: R+[0 , ~,[ is a continuous function having the following properties: