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Register a new userEigenfunction expansions of ultradifferentiable functions and ultradistributions. III. Hilbert spaces and Universality
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In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary.
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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 system. This generalizes Grothendiecks classical result that the space $mathcal{O}_M$ of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not.
We study functions of bounded variation (and sets of finite perimeter) on a convex open set $Omegasubseteq X$, $X$ being an infinite dimensional real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein--Uhlenbeck operator.
We prove an uncertainty principle for certain eigenfunction expansions on $ L^2(mathbb{R}^+,w(r)dr) $ and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on $ mathbb{C}^n $ and Hermite operator on $ mathbb{R}^n.$
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.
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 exponential weights. In this setting we define the $omega$-wave front set $WF_omega(u)$ and the Gabor $omega$-wave front set $WF^G_omega(u)$ of $uinmathcal{S}_{omega}$, and we prove that they coincide. Finally we look at applications of this wave front set for operators of differential and pseudo-differential type.
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