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Generalized Fourier Frames in Terms of Balayage

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Abstract

We develop a theory of non-uniform sampling in the context of the theory of frames for the settings of the short time fourier transform and pseudo-differential operators. Our theory is based on profound historical precedents including Beurling’s theory of balayage, emanating from the nineteenth century work of Christoffel and Poincaré, the theory and results from spectral synthesis due to Wiener and Beurling and a host of the major harmonic analysts of the twentieth century, and the theory of sets of multiplicity, going back to Riemann and emerging fundamentally from the Russian school of harmonic analysis in the early twentieth century. Our results are meant to serve as the underpinnings for both theoretical and practical results in the realm of non-uniform sampling. They can also be compared with several other distinct forays into non-uniform sampling, including the settings of quasi-crystals and modulation spaces, where proofs for the latter setting require the analysis of convolution operators on the Heisenberg group. Our theory herein is the first step in which the ultimate goal is computational implementation for non-uniform sampling and its myriad applications, where balayage, spectral synthesis, and sets of multiplicity are computationally quantified. A critical component is to resurrect the formulation of balayage in terms of covering criteria.

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Acknowledgments

The first named author gratefully acknowledges the support of MURI-AFOSR Grant FA9550-05-1-0443. The second named author gratefully acknowledges the support of MURI-ARO Grant W911NF-09-1-0383, NGA Grant HM-1582-08-1-0009, and DTRA Grant HDTRA 1-13-1-0015. Both authors benefited from insightful observations by Professors Carlos Cabrelli, Hans Feichtinger, Karlheinz Gröchenig, Matei Machedon, Basarab Matei, Ursula Molter, and Kasso Okoudjou, as well as important technical assistance for one of our examples by Professor Gröchenig. We have also incorporated the referees’ suggestions, that we very much appreciate. In general, these suggestions have translated into an adjusted and expanded introduction, as well as some added and suppressed details. Further, although the interest of the second named author in this topic goes back to the 1960s, he is especially appreciative of his collaboration in the late 1990s with Dr. Hui-Chuan  Wu related to Theorem 4.2 and the material in Sect. 6. Finally, the second named author has had the unbelievably good fortune through the years to learn from Henry J. Landau, a grand master in every way. His explicit contributions for this paper are noted in Sect. 4.2.

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  1. Department of Mathematical Sciences, DePaul University, Chicago, IL, 60614, USA

    Enrico Au-Yeung

  2. Norbert Wiener Center, Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    John J. Benedetto

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  1. Enrico Au-Yeung
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Correspondence to John J. Benedetto.

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Communicated by Hans G. Feichtinger.

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Au-Yeung, E., Benedetto, J.J. Generalized Fourier Frames in Terms of Balayage. J Fourier Anal Appl 21, 472–508 (2015). https://doi.org/10.1007/s00041-014-9369-7

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