Non harmonic Fourier series and its applications (Q1108523)

Consider a nonharmonic Fourier series \(f\sim \sum c_ ne^{-i\omega_ nt}(-T<t<T)\). Here \(\omega_ n\) are square roots of the eigenvalues of a Sturm-Liouville operator. We put \(\| c_ n\|_{2,b}=\{\sum (n\quad 2+1)\quad b| c_ n| \quad 2\}^{1/2}\) and \(\dot Y{}_ b=H\) \(b(-T,T)(b\geq 0),=H^{-b}(-T,T)\) * \((b<0)\) where H b(-T,T) denote the usual Sobolev space of order b and \(H^{-b}(-T,T)\) denotes the adjoint space of H b(-T,T). The paper discusses the following estimates: \(K_ 1\| f\|_{\dot Y_ b}\leq \| c_ n\|_{2,b}\leq K_ 2\| f\|_{\dot Y_ b}\). The problem of controllability for a one- dimensional vibrating system is considered as an application.