Density Results for Continuous Frames
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(2020) 26:56
Density Results for Continuous Frames Mishko Mitkovski1 · Aaron Ramirez2 Received: 14 September 2017 / Revised: 16 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We derive necessary conditions for localization of continuous frames in terms of generalized Beurling densities. As an important application we provide necessary density conditions for sampling, interpolation, and uniform minimality in a very large class of reproducing kernel Hilbert spaces. Keywords Frame · Continuous frame · Interpolation · Sampling · Uniformly minimal · Beurling densities 2000 Mathematics Subject Classification. 42C15 · 46E22 · 94A20
1 Introduction A well-known elementary linear algebra fact says that any linearly independent set of vectors in a finite-dimensional vector space cannot have more elements than any spanning set. In other words, the cardinality of any Riesz sequence cannot be greater than the cardinality of any frame. Even though there is no exact analog of these results in the infinite-dimensional setting there are many well-known results which are very similar in spirit. In the infinite-dimensional setting one needs to replace the comparison of cardinalities with a more suitable concept—which is the concept of densities. Basically one needs to compare the cardinalities locally everywhere and
Communicated by Massimo Fornasier. Mishko Mitkovski: Research supported in part by National Science Foundation DMS grant # 1600874.
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Mishko Mitkovski [email protected] Aaron Ramirez [email protected]
1
School of Mathematical and Statistical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, SC 29634, USA
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Escuela de Matematica, Universidad de El Salvador, Ciudad Universitaria, San Salvador, El Salvador 0123456789().: V,-vol
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then take the appropriate limits. The first density results were obtained in the context of non-harmonic complex exponentials. The first definitive results were proved by Beurling [7] and Kahane [18] who almost characterized frames and Riesz sequences of complex exponentials in terms of certain natural densities of their frequency sequence. These densities are now known as uniform densities or Beurling densities. These results were later extended and generalized in various ways and to many different settings [1,4,17,19,25,26,28–30,32]. The most important and popular approaches for proving the necessary part of density theorems are due to Landau [19], Ramanathan and Steger [26], Balan et al [5,6], and the recent one of Nitzan and Olevskii [23]. Most density results for sampling and interpolation pertain to a specific reproducing kernel Hilbert space (RKHS). In this paper we provide a universal density theorem which applies to a wide class of RKHSs and can be used as an alternative new method for proving density theorems. Besides being the most general, one additional advantage of our density result is that it allows comparison of two “truly” conti
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