Abstract
Matrix-type operators with the off-diagonal decay of polynomial or sub-exponential types are revisited with weaker assumptions concerning row or column estimates, still giving the continuity results for the frame type operators. Such results are extended from Banach to Fréchet spaces. Moreover, the localization of Fréchet frames is used for the frame expansions of tempered distributions and a class of Beurling ultradistributions.
Highlights
Localized frames were introduced independently by Gröchenig [23] and Balan, Casazza, Heil, and Landau [2,3]
The localization conditions in [23] are related to off-diagonal decay of the matrix determined by the inner products of the frame elements and the elements of a given Riesz basis
We have chosen to stick to the localization concept from [23], because the results obtained for a family of Banach spaces there can naturally be related to Fréchet frames
Summary
We consider polynomially and exponentially localized frames in the way defined in [23], and sub-exponential localization. – polynomially localized with respect to G with decay γ > 0 (in short, γ -localized wrt (gn)∞ n=1) if there is a constant Cγ > 0 so that max{| em, gn |, | em , gn |} ≤ Cγ (1 + |m − n|)−γ , m, n ∈ N;. – exponentially localized with respect to G if for some γ > 0 there is a constant Cγ > 0 so that max{| em , gn |, | em , gn |} ≤ Cγ e−γ |m−n|, m, n ∈ N. – β-sub-exponentially localized with respect to G (for β ∈ (0, 1)) if for some γ > 0 there is Cγ > 0 so that max{| em , gn |, | em , gn |} ≤ Cγ e−γ |m−n|β , m, n ∈ N
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
