Sampling theory and bases of exponentials: novel techniques in the foundations of communications

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Sampling theory is a vibrant area of mathematical investigation and it continues to play a central role in applied sciences and electrical engineering. In recent years, fundamental progress has been achieved in the classical sampling theory and the theory of exponential bases and frames. To this end, new constructive methods were developed and functional analytic insights such as those stemming from the groundbreaking resolution of the Kadison-Singer problem were applied.This project builds on these new results, which include newly obtained properties of Riesz bases and frames of exponentials on Paley-Wiener spaces on finite unions of intervals and/or on finite measure sets, and the lack of uniform convergence of the Nyquist reconstruction formula in Paley-Wiener spaces of functions with non-square-integrable Fourier transforms. We seek Riesz bases and frames of exponentials with additional properties and refine results that are based on the positive solution of the Kadison-Singer problem. We also seek to obtain constructive versions of the existence results that are available today. Furthermore, we extend results on the lack of uniform convergence of Nyquist reconstruction formulas to irregular sampling formulas, for example, those developed by Papoulis. We expect that multiple related research foci in this proposal will lead to significant cross fertilization.The questions considered herein address the foundations of communications engineering and are directly relevant for current tasks in data transmission. For example, the Balian-Low theorem for subspaces shows that avoiding critical density does not fully mitigate the Balian-Low phenomenon. This may have impact on the fifth generation mobile communications where time-frequency transmission methods such as orthogonal frequency division multiplexing are employed. We are guided by these considerations and hope to benefit from our collaboration with electrical engineers working in sampling theory and communications.