Operations And Isomorphism In Non-Quasianalytic Functions And Quasianalytic Functionals

Classes of ultradifferentiable functions have been used and investigated since the twenties of the last century. According to the theorem of Denjoy-Carleman they come in two groups:the quasianalytic and the non-quasianalytic classes. In the eight-ies of the last century, Meise,Bonet and Taylor etc introducedω—ultradifferentiable functions andω—ultradistributions by modifing the condition of the subadditivity ofωgiven by Beurling, Retzsche and replaced it by the weaker condition (a) (see definition 2.1). Since then the properties, operations and Fourier transform in these spaces have been discussed and been applied in the theory of linear partial differential operaters, and many significant results have been obtained.In this paper, we further discuss the multiplications and convolutions in theω—ultradifferentiable functions spaces D*(RN) and andω—ultradistributionsε'*(RN) of non-quasianalytic class, and the density and isomorphism in the space of quasiana-lytic functionalsε(ω)(G). We get the following conclusions.Theorem 1 Let w be a weight function of non-quasianalytic,f∈D(ω)(RN),g∈ε'(ω)(RN).Then we have f * g∈D(ω)(RN) and f*g= f.g.Theorem 2 For weight function of non-quasianalyticω, D*(Ω) is self-closed with continuous multiplication.Theorem 3 Letωbe a weight function of quasianalytic, letΩ(?) RN be open, and suppose that forμ∈ε'(ω)(Ω), there exeistλ, C> 0 and a compact K (?)Ωsuch that Thenμis entire and satisfiesTheorem 4 Let w be a weight function of of quasianalytic and let G be an open set in RN. Then H(CN) is dense inε(ω)(G).Theorem 5 For each weight function of quasianalyticωand each convex open set G (?) RN, the Fourier-Laplace transform F:ε'(ω)(G)â†'A(ω)(CN,G) is a linear topological isomorphism.