Convolutions And Fourier Transforms In ω-Ultradistributions

Classes of ultradifferentiable functions have been used and investigated since the twenties of the last century. Based on these, R.Meise, B.A.Taylor, D.Vogt and J.Bonet etc extended the theory of distributions, and introduced the conceptions ofω—ultra-differentiable functions spaces andω—ultradistributions by weight functions. In these spaces, they began the investigation to the theory of linear partial differential operators and obtained many significance results.Classes of non-quasianalytic functions are classically defined by imposing growth conditions on the derivatives of the functions. It was Beurling who pointed out that decay properties of the Fourier transform of a compactly supported function can be used for this purpose equally well. Therefore, Fourier transform become a important and useful method to investigate ultradifferentiable functions, ultradistributions and the theory of linear partial differential operators.In the paper, we discuss the multiplications and convolutions inω—ultradifferentiable functions spaces andω—ultradistributions by Fourier-Laplace transform based on the results of R.Meise, B.A.Taylor, D.Vogt and J.Bonet etc, and obtain the following:Theorem Transform Tμ:D(ω)(RN)�'D(ω)(RN) and sμ:ε'*(RN)�'ε'*(RN) is continuous. Theorem Let f∈D(ω),g∈D'(ω)(RN). Then we have f*g∈(ω)(RN), f*g＝f.g, and fg∈D(ω)(RN)...