Shrinking and boundedly complete Schauder frames in Fréchet spaces

Abstract

ElsevierBonet Solves, JA.; Fernandez, C.; Galbis, A.; Ribera Puchades, JM. (2014) AbstractWe study Schauder frames in Fréchet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete Schauder frames on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional Schauder frame is shrinking or boundedly complete in terms of properties of the space. Several examples of concrete Schauder frames in function spaces are also presented.

Localization of Fréchet frames and expansion of generalized functions

“…For another approach to frames in Fréchet spaces we refer to [5]. For more on frames for Banach spaces, see e.g.…”

Frames and representing systems in Fréchet spaces and their duals

“…Proof. (a) According to the proof of [7,Theorem 1.4] the operators U : E → Λ and S : Λ → E given by U(x) := (x ′ i (x)) i and S((α i ) i ) := ∞ i=1 α i x i respectively, are continuous and S • U = id E . Consequently (x ′ i ) i is a frame for E with respect to Λ.…”

“…On the one hand we study Λ-Bessel sequences (g i ) i ⊂ E ′ , Λ-frames and frames with respect to Λ in the dual of a Hausdorff locally convex space E, in particular for Fréchet spaces and complete (LB)-spaces E, with Λ a sequence space. We investigate the relation of these concepts with representing systems in the sense of Kadets and Korobeinik [12] and with the Schauder frames, that were investigated by the authors in [7]. On the other hand our article emphasizes the deep connection of frames for Fréchet and (LB)-spaces with the sufficient and weakly sufficient sets for weighted Fréchet and (LB)-spaces of holomorphic functions.…”

Riesz-Like Bases in Rigged Hilbert Spaces

“…There is a wide interest and a rich literature on bases or frames in locally convex spaces (in particular, Banach spaces) and on their existence, see, e.g. [16,25] and [14] and references therein.…”

“…Finally, the sequence {ξ n } is basis consisting of orthonormal vectors in D[ · +1 ]; indeed, (14) ξ i |ξ j +1 = T ξ i |T ξ j = e i |e j = δ ij , i, j ∈ N.…”