Fourier Sampling Numbers

• Fourier sampling numbers quantify the minimal number of samples needed to reconstruct a signal within a prescribed accuracy.
• They vary with function classes—such as bandlimited, sparse, Besov, and Barron—guiding optimal measurement and recovery strategies.
• These concepts underpin practical applications in signal processing and imaging, enabling robust recovery via both linear and nonlinear techniques.

Fourier sampling numbers quantify the minimal number of samples (typically in time or space) or measurements (typically in frequency) needed to reconstruct or approximate a signal or function from a given class to within prescribed accuracy. This concept is central to signal processing, harmonic analysis, approximation theory, and machine learning, particularly in determining optimal rates and strategies for function recovery under structural or smoothness constraints. The detailed characterization of Fourier sampling numbers varies according to the function class (e.g., bandlimited, Besov, Barron, or sparse), the reconstruction norm (e.g., $L^2$, $L^\infty$), the domain (continuous, periodic, discrete, or finite group), and the sampling model (uniform, non-uniform, adaptive, linear, or nonlinear).

1. Formal Definitions and General Frameworks

Let $\mathcal{C}$ be a class of functions (or signals) $f:\mathcal{D}\to\mathbb{C}$ with prescribed properties—e.g., bandlimited, $k$-Fourier-sparse, Besov smoothness, or membership in the Barron space. Given a family of permissible sampling functionals (time samples, Fourier coefficients, etc.), the Fourier sampling number $s_m(\mathcal{C}; X)$ or $s_N(\epsilon;\mathcal{C})$ is the minimal $m$ such that any $f\in\mathcal{C}$ can be reconstructed (in norm $X$) to error $L^\infty$0 from $L^\infty$1 samples or measurements. Throughout, the underlying recovery can be constrained to linear/nonlinear, deterministic/randomized, uniform/non-uniform, or even spectrum-blind strategies.

For example, for bandlimited or $L^\infty$2-Fourier-sparse signals on $L^\infty$3, $L^\infty$4 denotes the smallest $L^\infty$5 such that the signal can be interpolated from $L^\infty$6 (possibly randomized) time-domain samples to $L^\infty$7-error at most $L^\infty$8 (Song et al., 2022, Avron et al., 2018).

2. Concrete Characterizations in Signal Classes

Bandlimited, Multiband, and $L^\infty$9-Sparse Classes

For classes characterized by Fourier support, the sampling number admits concrete expressions:

• Bandlimited ($\mathcal{C}$0 on $\mathcal{C}$1): $\mathcal{C}$2, matching classical Nyquist rates up to logarithms, with efficient algorithms approaching these rates (Avron et al., 2018).
• Multiband: $\mathcal{C}$3, matching the Landau rate for unions of $\mathcal{C}$4 bands.
• $\mathcal{C}$5-Fourier-Sparse: For signals with $\mathcal{C}$6 nonzero Fourier components, $\mathcal{C}$7, and sample-optimal algorithms achieve $\mathcal{C}$8 nonuniform samples (Avron et al., 2018).
• Continuous $\mathcal{C}$9-Fourier-Sparse Interpolation: Recent work established the optimality of quartic sample complexity: $f:\mathcal{D}\to\mathbb{C}$0 for the minimal number of time samples needed for robust interpolation up to noise, closing the previous gap between information-theoretic lower and algorithmic upper bounds (Song et al., 2022).

Besov, Barron, and Decay-Smoothness Spaces

For broader smoothness spaces, the rates are determined by approximation theory:

• Besov Spaces $f:\mathcal{D}\to\mathbb{C}$1: For periodic functions on $f:\mathcal{D}\to\mathbb{C}$2, the Fourier sampling number $f:\mathcal{D}\to\mathbb{C}$3 admits the decay

$f:\mathcal{D}\to\mathbb{C}$4

in the "linear" parameter regime ($f:\mathcal{D}\to\mathbb{C}$5 or $f:\mathcal{D}\to\mathbb{C}$6), with matching lower and upper bounds up to polylogarithmic factors for $f:\mathcal{D}\to\mathbb{C}$7. In the "nonlinear" regime ($f:\mathcal{D}\to\mathbb{C}$8 or $f:\mathcal{D}\to\mathbb{C}$9), additional logarithmic penalties arise (Siegel, 19 Aug 2025). The results specify nearly optimal Fourier measurement schemes (low-pass in the linear regime, hierarchical random subsampling in the nonlinear regime) and constructive convex recovery.

• Fourier-Analytic Barron Space: For $k$0 with $k$1, the sampling number (optimal $k$2 to reach error $k$3 in $k$4) scales as

$k$5

up to logarithmic factors, demonstrating dimension-independent algebraic rates for nonlinear sampling, in contrast to the classical curse of dimensionality for linear schemes (Voigtlaender, 2022).

Finite Group and Discrete Settings

For functions on finite Abelian groups (e.g., $k$6), a universal sampling set $k$7 is a subset such that any $k$8-sparse Fourier signal can be reconstructed from its restriction to $k$9. Explicit constructions partition $s_m(\mathcal{C}; X)$0 into subspaces to guarantee "m-generating" properties, yielding

$s_m(\mathcal{C}; X)$1

with provable $s_m(\mathcal{C}; X)$2-error bounds. No sub-quadratic (in $s_m(\mathcal{C}; X)$3) lower bounds are known (Morotti, 2015).

3. Quantitative Error Bounds and Sampling Rate Asymptotics

General results relate the Fourier sampling number to smoothness and decay parameters:

• Continuous $s_m(\mathcal{C}; X)$4-Domain, Sobolev and Tail Conditions:

For $s_m(\mathcal{C}; X)$5 with smoothness $s_m(\mathcal{C}; X)$6 and spatial decay exponent $s_m(\mathcal{C}; X)$7, the root-mean-square error in DFT-based approaches satisfies

$s_m(\mathcal{C}; X)$8

The minimax optimal $s_m(\mathcal{C}; X)$9 for target error $s_N(\epsilon;\mathcal{C})$0 is

$s_N(\epsilon;\mathcal{C})$1

• pure Sobolev ($s_N(\epsilon;\mathcal{C})$2): $s_N(\epsilon;\mathcal{C})$3.
  • pure decay ($s_N(\epsilon;\mathcal{C})$4): $s_N(\epsilon;\mathcal{C})$5.
  • both decay and smoothness: $s_N(\epsilon;\mathcal{C})$6 for additional frequency decay $s_N(\epsilon;\mathcal{C})$7 (Ehler et al., 2024).
  • Uniform (ℓ$s_N(\epsilon;\mathcal{C})$8) Error Bounds:

Recent NFFT-based theory gives explicit uniform bounds for any $s_N(\epsilon;\mathcal{C})$9, e.g.,

$m$0

for polynomial decay $m$1, $m$2, with $m$3 scaling as $m$4 or $m$5 depending on the dominant term (Potts et al., 22 Jun 2026).

4. Algorithmic and Measurement Strategies

Spectrum-Aware Sampling and Convex Recovery

• For large classes (bandlimited, sparse, Gaussian process), universal non-uniform sampling—via ridge leverage score sampling—achieves (up to log factors) the minimal possible sample complexity for all $m$6 (the Fourier spectrum prior measure). The recovery is performed by kernel ridge regression in the continuous operator framework (Avron et al., 2018).
• Besov and Barron classes admit nearly optimal measurement strategies: low-pass frequency blocks for "linear" regimes, and scale-adapted random subsampling for "nonlinear" regimes, with reconstruction via convex minimization (e.g., $m$7 or Barron seminorm) (Siegel, 19 Aug 2025, Voigtlaender, 2022).

Sparse and Finite Group Algorithms

Explicit universal sets and combinatorial partitioning underlie efficient $m$8-bounded recovery for $m$9-sparse signals on $f\in\mathcal{C}$0 (Morotti, 2015).

5. Information-Theoretic Lower Bounds and Optimality Gaps

• For $f\in\mathcal{C}$1-Fourier-sparse continuous interpolation, the information-theoretic lower bound for sample complexity is $f\in\mathcal{C}$2, achieved algorithmically for the first time using high SNR bands and structural decomposition of signals (Song et al., 2022).
• In general signal classes, information-theoretic lower bounds show that the sample complexity cannot be asymptotically improved beyond the statistical dimension $f\in\mathcal{C}$3 up to logarithmic factors (Avron et al., 2018).
• For Besov norms with $f\in\mathcal{C}$4 and certain $f\in\mathcal{C}$5, an unavoidable polylogarithmic penalty separates the Fourier sampling numbers from Gelfand widths, demonstrating gaps in the nonlinear regime (Siegel, 19 Aug 2025).

6. Practical Implications and Applications

• Edge Recovery and Imaging: In bounded-variation images ($f\in\mathcal{C}$6), well-designed Fourier sampling coupled with $f\in\mathcal{C}$7-minimization achieves edge localization rates $f\in\mathcal{C}$8, outperforming grid-based sampling. Accurate reconstruction of sharp features is closely linked to the decay regime of the Fourier sampling number (Siegel, 19 Aug 2025).
• Efficient FFT-based Approximation: The NFFT-based framework provides explicit recipes for choosing sampling and truncation parameters to ensure prescribed $f\in\mathcal{C}$9 errors, with computational cost $X$0 (Potts et al., 22 Jun 2026).
• Universal Recovery in Signal Processing: Spectrum-blind sampling designs match Nyquist and Landau rates for a broad array of models, extend to kriging and Gaussian process regression in one dimension, and provide robust guarantees under noise (Avron et al., 2018).

7. Comparative Table of Fourier Sampling Rates

Function Class	Sampling Number $X$1	Key Parameters	Reference
$X$2-Fourier-sparse (continuous)	$X$3	$X$4: sparsity, $X$5: bandlimit	(Song et al., 2022)
Bandlimited $X$6	$X$7	$X$8: max frequency, $X$9: interval	(Avron et al., 2018)
Barron space $L^\infty$00	$L^\infty$01	$L^\infty$02: smoothness, $L^\infty$03: dim.	(Voigtlaender, 2022)
Besov $L^\infty$04	$L^\infty$05 (linear regime)	$L^\infty$06: smoothness, $L^\infty$07: dim., $L^\infty$08	(Siegel, 19 Aug 2025)
Sobolev $L^\infty$09	$L^\infty$10	$L^\infty$11: smoothness order	(Ehler et al., 2024)
Polynomial decay (no smoothness)	$L^\infty$12	$L^\infty$13: decay exponent	(Ehler et al., 2024)
$L^\infty$14-sparse (finite group)	$L^\infty$15	$L^\infty$16: field size, $L^\infty$17: dim., $L^\infty$18	(Morotti, 2015)

These rates are optimal (up to polylogarithmic factors) under the stated assumptions.

8. Open Problems and Directions

• Removal of logarithmic factors (e.g., in Barron and Besov regimes), closing the remaining gaps to information-theoretic lower bounds (Voigtlaender, 2022, Siegel, 19 Aug 2025).
• Random versus deterministic sampling: it remains open whether simple random sampling can always achieve the optimal rates guaranteed by sophisticated leverage-score or combinatorial constructions (Voigtlaender, 2022).
• Precise constant dependence on dimension, structure of function class, and transition regions between linear and nonlinear measurement regimes.
• Extension of these results to function spaces beyond Barron/Besov/Sobolev, such as deep Barron-type or kernel-based classes, or to pointwise $L^\infty$19 recovery in high dimension (Voigtlaender, 2022, Siegel, 19 Aug 2025).

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References: (Song et al., 2022, Avron et al., 2018, Voigtlaender, 2022, Morotti, 2015, Ehler et al., 2024, Siegel, 19 Aug 2025, Potts et al., 22 Jun 2026)