FOURIER COEFFICIENTS OF PERIODIC FUNCTIONS OF GEVREY CLASSES AND ULTRADISTRIBUTIONS
Yoshiko Taguchi · Institutional Repositories DataBase (IRDB) · 1987
\S 0. Introduction.Gevrey classes of ultradifferentiable functions on an open domain $\Omega\subset R$ " of type $s(s>1)$ and their dual spaces are extensively studied by Professor H. Komatsu ([1], [2]).In the case of one point support, J. P. Ramis ([7], [8]) has studied Gevrey classes of functions of type $s(-\infty0$ there exists $C_{A}>0$ such that $\Vert f^{(n)}\Vert_{\infty}\leq C_{A}\cdot A^{n}\cdot(n!)^{\epsilon}$ for any integer $n\geq 0$ .$f\in C_{(\$)}(T)$ is called a ultradifferentiable function of Gevrey- Beurling class of type $s$ .)$ be real valued functions on $x>0,$ $r>0$ respectively.Then $F(x)$ and $G(r)$ are associated if they are connected by the following formulas: $\inf_{x>0}F(x)\cdot r^{-x}=G(r)$ and $\sup_{r>0}G(r)\cdot r^{x}=F(x)$ .$W$ call $(F(x), G(r))$ an associated pajr.Diffinition 2.2.Let $(F(x), G(r)),$ ( $p_{(x)},$ $G_{(r))}$ be two associated pairs, then we call them equivalent if there exists $A>0$ such that $F(x)=A^{x}\cdot F(x)$ for all $x>0$ and $\tilde{G}(r)=G(\frac{r}{A})$ for all $r>0$ .