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2021 ◽  
Vol 13 ◽  
Author(s):  
Pavol Jan Zlatos

Using the ideas of E. I. Gordon we present and farther advancean approach, based on nonstandard analysis, to simultaneousapproximations of locally compact abelian groups and their dualsby (hyper)finite abelian groups, as well as to approximations ofvarious types of Fourier transforms on them by the discrete Fouriertransform. Combining some methods of nonstandard analysis andadditive combinatorics we prove the three Gordon's Conjectureswhich were open since 1991 and are crucial both in the formulationsand proofs of the LCA groups and Fourier transform approximationtheorems


Author(s):  
Abdolreza Tahmasebi Birgani ◽  
Mohammad Sadegh Asgari
Keyword(s):  

2020 ◽  
Vol 54 (2) ◽  
pp. 211-219
Author(s):  
S.Yu. Favorov

We show that if points of supports of two discrete ”not very thick” Fourier transformable measures on locally compact abelian (LCA) groups tend to one another at infinity and the same is true for the masses at these points, then these measures coincide. The result is valid for discrete almost periodic measures on LCA groups too. Also, we show that the result is false for some discrete ”thick” measures. To do this, we construct a discrete almost periodic measure on the real axis, whose masses at the points of support tend to zero as these points approach infinity.


Author(s):  
Divya Jindal ◽  
Uttam Kumar Sinha ◽  
Geetika Verma

In this paper, we study multivariate Gabor frames in matrix-valued signal spaces over locally compact abelian (LCA) groups, where the lower frame condition depends on a bounded linear operator [Formula: see text] on the underlying matrix-valued signal space. This type of Gabor frame is also known as a multivariate [Formula: see text]-Gabor frame. By extending work of Gǎvruta, we present necessary and sufficient conditions for the existence of [Formula: see text]-Gabor frames of multivariate matrix-valued Gabor systems. Some operators which can transform multivariate matrix-valued Gabor and [Formula: see text]-Gabor frames into [Formula: see text]-Gabor frames in terms of adjointable operators are discussed. Finally, we give a Paley–Wiener-type perturbation result for multivariate matrix-valued [Formula: see text]-Gabor frames.


Author(s):  
O O Enwo ◽  
E Player ◽  
N Steel ◽  
J A Ford

ABSTRACT Background Inequalities in life events can lead to inequalities in older age. This research aimed to explore associations between life events reported by older people and quality of life (QoL) and functional ability. Methods Participants were grouped according to eight life events: parental closeness, educational opportunities in childhood, financial hardship, loss of an unborn child, bereavement due to war, involvement in conflict, violence and experiencing a natural disaster. Linear and logistic regressions were used to explore associations between these groups and the main outcomes of functional ability and QoL. Results 7555 participants were allocated to four LCA groups: ‘few life events’ (n = 6,250), ‘emotionally cold mother’ (n = 724), ‘violence in combat’ (n = 274) and ‘many life events’ (n = 307). Reduced QoL was reported in the ‘many life events’ (coefficient − 5.33, 95%CI −6.61 to −4.05), ‘emotionally cold mother’ (−1.89, −2.62 to 1.15) and ‘violence in combat’ (−1.95, −3.08 to −0.82) groups, compared to the ‘few life events’ group. The ‘many life events’ group also reported more difficulty with activities of daily living. Conclusions Policies aimed at reducing inequalities in older age should consider events across the life course.


Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 25 ◽  
Author(s):  
Hans G. Feichtinger

The Banach Gelfand Triple ( S 0 , L 2 , S 0 ′ ) ( R d ) consists of S 0 ( R d ) , ∥ · ∥ S 0 , a very specific Segal algebra as algebra of test functions, the Hilbert space L 2 ( R d ) , ∥ · ∥ 2 and the dual space S 0 ′ ( R d ) , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce S 0 ( R d ) , ∥ · ∥ S 0 and hence ( S 0 ′ ( R d ) , ∥ · ∥ S 0 ′ ) , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra S 0 ( R d ) , ∥ · ∥ S 0 can be used to establish this natural identification.


2019 ◽  
Vol 266 ◽  
pp. 106843
Author(s):  
Dekui Peng ◽  
Wei He
Keyword(s):  

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