We study the nuclearity of the Gelfand-Shilov spaces S (M) (W) and S {M} {W } , defined via a weight (multi-)sequence system M and a weight function system W. We obtain characterizations of nuclearity for these function spaces that are counterparts of those for Köthe sequence spaces. As an application, we prove new kernel theorems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces S (M) (A) and S {M} {A} (defined via weight sequences M and A) and the Beurling-Björck spaces S (ω) (η) and S {ω} {η} (defined via weight functions ω and η). Our results cover anisotropic cases as well.

We characterize the nuclearity of the Beurling-Björck spaces S (ω) (η) (R d ) and S {ω} {η} (R d ) in terms of the defining weight functions ω and η.A weight function ω is called radially increasing if ω(x) ≤ ω(y) whenever |x| ≤ |y|. Given a weight function ω and a parameter λ > 0, we introduce the family of norms ϕ ω,λ = sup x∈R d |ϕ(x)|e λω(x) .

We study weighted (P LB)-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 Grothendieck's classical result that the space 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.2010 Mathematics Subject Classification. 46E10, 46A08, 46A13, 46A63. Key words and phrases. Ultrabornological (PLB)-spaces; Gelfand-Shilov spaces; multiplier spaces; short-time Fourier transform; Gabor frames.A. Debrouwere was supported by FWO-Vlaanderen through the postdoctoral grant 12T0519N. L. Neyt gratefully acknowledges support by FWO-Vlaanderen through the postdoctoral grant 12ZG921N.

We use methods from time-frequency analysis to study the structural and linear topological properties of the space 9 B 1ω of ultradistributions vanishing at infinity (with respect to a weight function ω). Particularly, we show the first structure theorem for 9 B 1ω under weaker hypotheses than were known so far. As an application, we determine the structure of the S-asymptotic behavior of ultradistributions.

We obtain structural theorems for the so-called S-asymptotic and quasiasymptotic boundedness of ultradistributions. Using these results, we then analyze the moment asymptotic expansion (MAE), providing a full characterization of those ultradistributions satisfying this asymptotic formula in the one-dimensional case. We also introduce and study a uniform variant of the MAE. Some of our arguments rely on properties of the short-time Fourier transform (STFT). We develop here a new framework for the STFT on various ultradistribution spaces.