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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 transform and give a full characterization in terms of Fourier and Wigner transforms for several variables of a Paley-Wiener theorem in this general setting, which is new in the literature. We also analyze this type of results when the support of the function is not compact using polynomials. Some examples are given.
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
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
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 analogu