2020/12/21 by Hans G. Feichtinger, Feichtinger, Hans G., Anupam Gumber +1
Computer Science · Mathematics · #40E05 #41A65 #43A10 #46A40 #46B50 #46F05 #46H25 #Abelian and tauberian theorems #Abelian group #Advanced Mathematical Modeling in Engineering #Banach space #Completeness (order theory) #Discrete mathematics #FOS: Mathematics #Functional Analysis (math.FA) #Generalization #Image and Signal Denoising Methods #Invariant (physics) #Linear subspace #Locally compact space #Mathematical Analysis and Transform Methods #Mathematical analysis #Mathematics #Primary 43A15 #Pure mathematics #Secondary 43A25 #math.FA #msc:40E05 #msc:41A65 #msc:43A10 #msc:43A15 #msc:43A25 #msc:46A40 #msc:46B50 #msc:46F05 #msc:46H25
paper · pdf · doi:10.48550/arxiv.2012.11127
revision of version 1, version 1 title modified in version 2, 15 pages
openalex publication_date 2020/12/21 · arxiv created 2022/03/19 · arxiv updated 2022/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The main result of this paper is a far reaching generalization of the completeness result given by V.~Katsnelson in a recent paper [35]. Instead of just using a collection of dilated Gaussians it is shown that the key steps of an earlier paper [27] by the authors, combined with the use of Tauberian conditions (i.e. the non-vanishing of the Fourier transform) allow us to show that the linear span of the translates of a single function g ∈ \boldsymbol\mathcal S(ℝd) is a dense subspace of any Banach space satisfying certain double invariance properties. In fact, a much stronger statement is presented: for a given compact subset M in such a Banach space (\boldsymbol B, ‖ ⋅ ‖\boldsymbol B) one can construct a finite rank operator, whose range is contained in the linear span of finitely many translates of g, and which approximates the identity operator over M up to a given level of precision. The setting of tempered distributions allows to reduce the technical arguments to methods which are widely used in Fourier Analysis. The extension to non-quasi-analytic weights respectively locally compact Abelian groups is left to a forthcoming paper, which will be technically much more involved and uses different ingredients.