Discretization, sampling, and the Fourier ratio
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| Format: | Preprint |
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2026
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| author | Iosevich, A. Palsson, E. Yavicoli, A. |
| author_facet | Iosevich, A. Palsson, E. Yavicoli, A. |
| contents | We derive fundamental sampling bounds for smooth signals in continuous settings without sparsity assumptions. By introducing the Fourier ratio as a measure of spectral compressibility induced by smoothness, we obtain explicit, deterministic bounds linking signal regularity to recoverability from incomplete random samples. For functions in $C^{2}([0,1]^{2})$ sampled on an $N$ by $N$ grid, we show that a random subset of spatial samples of size $$ C\frac{r_{N}^{2}}{\eps^{2}}\log(r_{N}/\eps)^{2}\log(N^{2}) $$ suffices, with high probability, to recover the entire discretized signal via $\ell^{1}$ minimization with relative $L^{2}$ error $O(\eps)$. We develop a parallel theory for bandlimited functions on the unit sphere, obtaining analogous recovery guarantees with sample complexity scaling polylogarithmically in the bandwidth. Our results establish smoothness as a deterministic prior that enforces compressibility in the Fourier domain, bridging continuous harmonic analysis with discrete compressed sensing in a unified information-theoretic framework. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_17493 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Discretization, sampling, and the Fourier ratio Iosevich, A. Palsson, E. Yavicoli, A. Classical Analysis and ODEs Functional Analysis 42A10, 94A12, 42C10, 33C55, 94A20, 90C25 We derive fundamental sampling bounds for smooth signals in continuous settings without sparsity assumptions. By introducing the Fourier ratio as a measure of spectral compressibility induced by smoothness, we obtain explicit, deterministic bounds linking signal regularity to recoverability from incomplete random samples. For functions in $C^{2}([0,1]^{2})$ sampled on an $N$ by $N$ grid, we show that a random subset of spatial samples of size $$ C\frac{r_{N}^{2}}{\eps^{2}}\log(r_{N}/\eps)^{2}\log(N^{2}) $$ suffices, with high probability, to recover the entire discretized signal via $\ell^{1}$ minimization with relative $L^{2}$ error $O(\eps)$. We develop a parallel theory for bandlimited functions on the unit sphere, obtaining analogous recovery guarantees with sample complexity scaling polylogarithmically in the bandwidth. Our results establish smoothness as a deterministic prior that enforces compressibility in the Fourier domain, bridging continuous harmonic analysis with discrete compressed sensing in a unified information-theoretic framework. |
| title | Discretization, sampling, and the Fourier ratio |
| topic | Classical Analysis and ODEs Functional Analysis 42A10, 94A12, 42C10, 33C55, 94A20, 90C25 |
| url | https://arxiv.org/abs/2601.17493 |