English

The Fourier Ratio and complexity of signals

Classical Analysis and ODEs 2025-11-26 v1 Information Theory math.IT

Abstract

We study the Fourier ratio of a signal f:ZNCf:\mathbb Z_N\to\mathbb C, FR(f) := Nf^L1(μ)f^L2(μ) = f^1f^2, \mathrm{FR}(f)\ :=\ \sqrt{N}\,\frac{\|\widehat f\|_{L^1(\mu)}}{\|\widehat f\|_{L^2(\mu)}} \ =\ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}, as a simple scalar parameter governing Fourier-side complexity, structure, and learnability. Using the Bourgain--Talagrand theory of random subsets of orthonormal systems, we show that signals concentrated on generic sparse sets necessarily have large Fourier ratio, while small FR(f)\mathrm{FR}(f) forces ff to be well-approximated in both L2L^2 and LL^\infty by low-degree trigonometric polynomials. Quantitatively, the class {f:FR(f)r}\{f:\mathrm{FR}(f)\le r\} admits degree O(r2)O(r^2) L2L^2-approximants, which we use to prove that small Fourier ratio implies small algorithmic rate--distortion, a stable refinement of Kolmogorov complexity.

Keywords

Cite

@article{arxiv.2511.19560,
  title  = {The Fourier Ratio and complexity of signals},
  author = {K. Aldaleh and W. Burstein and G. Garza and G. Hart and A. Iosevich and J. Iosevich and A. Khalil and J. King and N. Kulkarni and T. Le and I. Li and A. Mayeli and B. McDonald and K. Nguyen and N. Shaffer},
  journal= {arXiv preprint arXiv:2511.19560},
  year   = {2025}
}