Exponential bases with integer frequencies

• Applicants: Andrei Caragea, Ph.D.; Professor Götz Eduard Pfander, Ph.D.
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 554877617

The foundational result in Fourier analysis states that the set of complex exponentials with integer frequencies forms an orthonormal basis for the space of square integrable functions on the unit interval, in short, the unit interval and the integers form an orthonormal spectral pair. Finding frequency sets that lead to bases for other domains is a key task in applied mathematics, engineering and the natural sciences. Aside of trivial extensions, for example using rescaling or a tensor argument for domains in higher dimensions, the task generally requires dropping the pairwise orthogonality of basis elements and, therefore, considering Riesz bases and frames as relaxations. In the past two decades, landmark results have been obtained in this area. For example, it was shown that for every finite union of intervals there exists a Riesz basis of exponentials, and a first set was constructed for which there exist no Riesz spectra. In addition, both directions of Fuglede's conjecture were shown to be wrong. The task of finding orthogonal or Riesz bases for a given domain becomes even more challenging if not all frequencies can be chosen for the task. This can arise for, example, due to to hardware limitations in electrical engineering and telecommunications applications or due to restrictions when modeling physical phenomena. Such tasks are generally equivalent to finding orthogonal or Riesz bases consisting of only integer frequencies for rescaled domains. Consequently, this project addresses the question of which subset of the unit interval, respectively of the d-dimensional unit cube, possess orthonormal bases, Riesz bases, or frames with frequencies contained in the integers. Central to our discussion are so-called hierarchical exponential bases for partitions of the unit interval or of the d-dimensional unit cube. That is, given a partition with sets indexed by a set K, does there exist a family of corresponding frequency sets so that for any subset J of K, the union of the frequency sets indexed by J forms a Riesz basis for the respective union of partitioning sets. One of the main technical tools that we plan to develop towards this goal is an extension of the classical result of Chebotarev that states that square submatrices of Fourier matrices of prime size are always invertible. Establishing analogues of this result in the case of Fourier matrices with non-prime size is a research direction of stand-alone interest, with applications in algebra and number theory.

• DFG Programme: Research Grants