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. K E Y W O R D S completeness of inductive limits, convolution, Gelfand-Shilov spaces, short-time Fourier transform, ultrabornological (PLS)-spaces M S C ( 2 0 1 0 ) 46A13, 46E10, 46F05, 46F10

Translation–Modulation Invariant Banach Spaces of Ultradistributions

“…, it is trivial that ϕ n (χ n * g) ∈ S * † (R d ) (this follows by direct verification, but, alternatively, it also follows from general results such as [5,Theorems 4.19 and 4.20]). The last part then follows by duality and the first part of the corollary.…”

Asymptotic boundedness and moment asymptotic expansion in ultradistribution spaces

“…The present section is devoted to developing a framework for the STFT of general ultradistributions with respect to compactly supported ultradifferentiable window functions of Beurling type. Our analysis closely follows that from [4, Section 2] (see also [3,Subsection 2.3]), though a bit more precision will be needed. Throughout this section we will always assume M p satisfies pM.1q, pM.2q 1 , and pM.3q 1 .…”

On the space of ultradistributions vanishing at infinity

“…We now study the mapping properties of the STFT on the spaces S pMpq pApq pR d q and S 1pMpq pApq pR d q (cf. [6]). We need some preparation.…”