On weighted inductive limits of spaces of ultradifferentiable functions and their duals

Abstract

In the first part of this paper we discuss the completeness of two general classes of weighted inductive limits of spaces of ultradifferentiable functions. In the second part we study their duals and characterize these spaces in terms of the growth of convolution averages of their elements. This characterization gives a canonical way to define a locally convex topology on these spaces and we give necessary and sufficient conditions for them to be ultrabornological. In particular, our results apply to spaces of convolutors for Gelfand-Shilov spaces.

arXiv comment35 pages

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• A projective description of generalized Gelfand-Shilov spaces of Roumieu type
• On the space of ultradistributions vanishing at infinity
• Convolutors of translation-modulation invariant Banach spaces of ultradistributions
• A multidimensional Tauberian theorem for Laplace transforms of ultradistributions
• Multiresolution expansions and wavelets in Gelfand-Shilov spaces
• Kernel theorems for Beurling-Björck type spaces
• Extension of Localisation Operators to Ultradistributional Symbols With Super-Exponential Growth