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We provide a projective description of the space $mathcal{E}^{{mathfrak{M}}}(Omega)$ of ultradifferentiable functions of Roumieu type, where $Omega$ is an arbitrary open set in $mathbb{R}^d$ and $mathfrak{M}$ is a weight matrix satisfying the analogue of Komatsus condition $(M.2)$. In particular, we obtain in a unified way projective descriptions of ultradifferentiable classes defined via a single weight sequence (Denjoy-Carleman approach) and via a weight function (Braun-Meise-Taylor approach) under considerably weaker assumptions than in earli
We study weighted $(PLB)$-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight syst
Given a non-quasianalytic subadditive weight function $omega$ we consider the weighted Schwartz space $mathcal{S}_omega$ and the short-time Fourier transform on $mathcal{S}_omega$, $mathcal{S}_omega$ and on the related modulation spaces with exponent
We develop real Paley-Wiener theorems for classes ${mathcal S}_omega$ of ultradifferentiable functions and related $L^{p}$-spaces in the spirit of Bang and Andersen for the Schwartz class. We introduce results of this type for the so-called Gabor tra
Given two systems $P=(P_j(D))_{j=1}^N$ and $Q=(Q_j(D))_{j=1}^M$ of linear partial differential operators with constant coefficients, we consider the spaces ${mathcal E}_omega^P$ and ${mathcal E}_omega^Q$ of $omega$-ultradifferentiable functions with
We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending previous wor