Eigenfunction expansions of ultradifferentiable functions and ultradistributions (Q2822707)

The authors study ultradifferentiable functions as well as the corresponding ultradistributions on a compact manifold \(X\). Given a positive elliptic (pseudo-)differential operator \(E\) of order \(\nu\in\mathbb{Z}_+\) on \(X\), they define \(\Gamma_{\{M_p\}}(X)\), the ultradifferentiable functions of Roumieu class, to be the space of all \(\varphi\in C^{\infty}(X)\) which satisfy NEWLINE\[NEWLINE\|E^k \varphi\|_{L^2(X)}\leq C h^{\nu k}M_{\nu k},\,\, k=0,1,2,\dots,NEWLINE\]NEWLINE for some \(C=C(\varphi)>0\) and \(h=h(\varphi)>0\), where \(M_p\), \(p\in\mathbb{N}\), is a sequence of positive numbers which satisfies some of the conditions of Komatsu; it is important to stress that the authors do not assume the sequence to be non-quasi-analytic, i.e., \(M_p\) can be \(p!\) which gives the analytic case. Of course, when \(M=p!^s\), \(s\geq 1\), this is nothing else but the Gevrey class. If the estimate is valid for every \(h>0\) and some \(C=C(h,\varphi)\), it gives the ultradifferentiable functions of Beurling class \(\Gamma_{(M_p)}(X)\).NEWLINENEWLINEThe authors prove that the definition is independent of the choice of \(E\). The advantage of this definition is that one can define \(\Gamma_{\{M_p\}}(X)\) without using local coordinates, i.e., the manifold needs to be `only' smooth. Furthermore, the authors prove that when \(X\) and \(E\) are analytic, \(\Gamma_{\{M_p\}}(X)\) is preserved by analytic change of variables and its elements are locally of \(\{M_p\}\) class. The space \(\Gamma_{\{M_p\}}(X)\) carries a natural topology and its dual is the space of ultradistributions \(\Gamma_{\{M_p\}}'(X)\).NEWLINENEWLINEThe main results of the article are characterisations of the elements of \(\Gamma_{\{M_p\}}(X)\) and \(\Gamma_{\{M_p\}}'(X)\) in terms of the eigenfunction expansions of the elliptic operator \(E\), i.e., the authors give necessary and sufficient conditions on the growth of \((f,\phi_j)\), \(j\in\mathbb{N}\), for \(f=\sum_j (f,\phi_j)\phi_j\) to be in \(\Gamma_{\{M_p\}}(X)\) as well as necessary and sufficient conditions for \(f\) to be in \(\Gamma_{\{M_p\}}'(X)\); here, \(\phi_j\), \(j\in\mathbb{N}\), are the eigenfunctions of \(E\).