Structure theorem for functionals in the space \(\mathfrak S^{\prime}_{\omega_1,\omega_{2}}\) (Q938487)

Summary: We introduce the space \({\mathfrak S}_{\omega_1\omega_2}\) of all \(C^\infty\) functions \(\varphi\) such that \(\sup_{|\alpha|\leq m}\|e^{k\omega_1} \partial^\alpha\varphi\|_\infty\) and \(\sup_{|\alpha|\leq m}\|e^{k\omega_2}\partial^\alpha \widehat{\varphi}\|_\infty\) are finite for all \(k\in\mathbb N_0\), \(\alpha\in\mathbb N_0^n\), where \(\omega_1\) and \(\omega_2\) are two weights satisfying the classical Beurling conditions. Moreover, we give a topological characterization of the space \({\mathfrak S}_{\omega_1\omega_2}\) without conditions on the derivatives. For functionals in the dual space \({\mathfrak S}_{\omega_1\omega_2}'\), we prove a structure theorem by using the classical Riesz representation theorem.