Band-Limited Coordinate Neural Network

• Band-limited coordinate neural networks are models that map continuous coordinates to signals while explicitly restricting the frequency spectrum for enhanced smoothness and anti-aliasing.
• They employ varied methods—such as frozen sine filters, interpolation-based low-pass reconstruction, and activation spectral shaping—to control effective bandwidth and support multiscale representation.
• Theoretical analyses provide sampling error bounds and trade-offs, highlighting benefits in interpolation accuracy and manufacturability while noting potential limits on high-frequency detail.

A band-limited coordinate neural network is a coordinate-based neural function—typically an MLP or neural field that maps continuous coordinates to signal values—whose spectrum, effective frequency support, or sampling behavior is explicitly constrained. In the literature, this constraint appears in several distinct forms: analytically band-limited coordinate architectures such as BACON, sampling-aware low-pass neural fields such as BANF, activation-level spectral shaping such as BandRC, topology-optimization parametrizations whose bandwidth encodes gradation limits, and shallow models whose hidden-parameter support is band-limited in ridgelet space (Lindell et al., 2021, Shabanov et al., 2024, Thennakoon et al., 16 May 2025, Chandrasekhar et al., 14 Aug 2025, Li et al., 2020). A complementary theoretical line analyzes coordinate MLPs trained on band-limited targets through sampling theory, proving smoothness, spectral decay, and explicit interpolation error bounds on finite domains (Giryes, 2020).

1. Definitions and formal scope

Coordinate networks, also called implicit neural representations, map continuous coordinates to signal values. In the notation used for INRs, a continuous target signal $S_x : \mathbb{R}^r \to \mathbb{R}^c$ is approximated by an INR $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$ trained by minimizing

$$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$

In BACON, the same general object is described as a coordinate-based network mapping $x \in \mathbb{R}^D$ to signal values $y(x)$; in TOBACO, the network maps coordinates to a composition vector $\vec\rho(x)$; and in BANF, the neural field is a continuous function later sampled on regular grids for downstream processing (Thennakoon et al., 16 May 2025, Lindell et al., 2021, Chandrasekhar et al., 14 Aug 2025, Shabanov et al., 2024).

The phrase band-limited is used for different objects in this literature. For target functions and sampling theory, it refers to the Fourier support of the signal itself. On the finite domain $[-\pi,\pi]^d$, the Fourier coefficients are written as

$$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$

and band-limited means there exists $K$ such that $c_k=0$ if there exists an index $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$0 with $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$1, so that

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$2

This periodic finite-domain formulation underlies the sampling-theoretic analysis of coordinate MLP interpolation (Giryes, 2020).

For architecture design, band limitation may instead be imposed by construction. BACON freezes sinusoidal frequencies so that the support of the output spectrum is bounded by a cumulative cutoff, while BANF inserts an interpolation-based reconstruction operator into the loss so that the optimized field is low-pass filtered relative to a chosen grid and kernel. BandRC shapes the frequency response at the activation level by using a raised-cosine impulse response with a learnable band shift, and TOBACO restricts the frequencies of its multiplicative filter network so that the represented composition field satisfies manufacturing gradation limits through Bernstein’s inequality (Lindell et al., 2021, Shabanov et al., 2024, Thennakoon et al., 16 May 2025, Chandrasekhar et al., 14 Aug 2025).

A further, non-equivalent notion appears in shallow random-feature theory. There, a depth-2 network is called band-limited when the hidden parameters $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$3 are restricted to a bounded parameter set such as

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$4

In that setting, band limitation constrains the accessible ridgelet spectrum rather than the ordinary Fourier spectrum of the output (Li et al., 2020).

2. Function-space characterization and sampling-theoretic analysis

A central theoretical treatment studies coordinate-based MLPs trained to interpolate data generated by smooth, spectrally constrained mappings through the lens of sampling theory. On the bounded domain $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$5, a feed-forward network with $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$6 layers is written recursively as

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$7

For non-expansive activations and bounded weights, the Jacobian is bounded by the product of layer norms:

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$8

Combining this with finiteness of the input domain and finitely many activation-induced derivative discontinuities yields finite total variation of the second derivative in one dimension, or of the Laplacian in multiple dimensions. The resulting corollary states that for finite multi-layer networks with bounded weights and non-expansive nonlinearities having finitely many discontinuities in their first derivative, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$9 is finite (Giryes, 2020).

This smoothness produces spectral decay. For univariate finite multi-layer networks under the same assumptions, the Fourier coefficients $$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$0 of the network output obey

$$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$1

For multivariate infinite-width two-layer ReLU networks with bounded weights, the corresponding decay is

$$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$2

for any $$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$3. A crucial point is that the neural interpolant need not itself be band-limited; the analysis instead uses the fact that its high-frequency content decays sufficiently fast to make sampling-theoretic error control possible (Giryes, 2020).

The same work derives explicit approximation bounds when the ground-truth mapping is band-limited and the network interpolates the sampled data. Under deterministic uniform sampling with $$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$4 in one dimension,

$$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$5

Under random sampling, if the sampling operator $$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$6 is invertible with condition number $$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$7,

$$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$8

with high probability. The analysis emphasizes the advantage of deterministic uniform sampling because orthogonal DFT inversion avoids the conditioning penalty. For multivariate uniform sampling on a grid, the bound becomes

$$L = \mathbb{E}_{x \in X}\,\|f_\theta(x)-\hat S_x\|^2.$$9

which approaches $x \in \mathbb{R}^D$0 in squared error for large $x \in \mathbb{R}^D$1 (Giryes, 2020).

These bounds are tied to classical sampling operators. On a uniform grid with $x \in \mathbb{R}^D$2 samples, one has $x \in \mathbb{R}^D$3 and exact coefficient recovery $x \in \mathbb{R}^D$4, with $x \in \mathbb{R}^D$5. For non-uniform samples, the system is $x \in \mathbb{R}^D$6 and the stability of $x \in \mathbb{R}^D$7 depends on the conditioning of $x \in \mathbb{R}^D$8. The paper makes the Shannon/Nyquist-type requirement explicit as $x \in \mathbb{R}^D$9 for exact reconstruction of a band-limited target, then quantifies the error incurred when the interpolating network is smooth but not strictly band-limited (Giryes, 2020).

3. Architectural mechanisms for enforcing or exploiting band limitation

The principal architectural families differ in where the band limitation is imposed: on frozen sinusoidal bases, on the training readout, on the activation response, or on an application-specific MFN parametrization. The following constructions are representative (Lindell et al., 2021, Shabanov et al., 2024, Thennakoon et al., 16 May 2025, Chandrasekhar et al., 14 Aug 2025).

Construction	Band-limiting mechanism	Noted properties
BACON	Frozen sine filters with per-layer bounds $y(x)$0	Analytical Fourier spectrum; multiscale outputs
BANF	Interpolation readout from regular samples with kernel $y(x)$1	Low-pass neural fields; anti-aliased LOD
BandRC	Raised-cosine activation with learnable $y(x)$2 and $y(x)$3	Compact spectral support with band shift
TOBACO	MFN frequencies $y(x)$4 with $y(x)$5	Implicit gradation limits via Bernstein inequality

BACON is built on Multiplicative Filter Networks. With input $y(x)$6, hidden width $y(x)$7, and sine filters

$y(x)$8

the forward pass is

$y(x)$9

The matrices $\vec\rho(x)$0 are frozen with entries sampled uniformly in $\vec\rho(x)$1. Because products of sines generate sums and differences of frequencies, the maximum frequency at output $\vec\rho(x)$2 is bounded by $\vec\rho(x)$3, and the overall bandwidth is $\vec\rho(x)$4. BACON therefore produces an analytically characterizable discrete Fourier spectrum on a finite periodic domain, and its outputs can be written as sums of sinusoids with the number of sine terms scaling as

$\vec\rho(x)$5

This makes the architecture explicitly interpretable in frequency space (Lindell et al., 2021).

BANF achieves band limitation without changing the backbone neural field. Instead, it inserts a reconstruction operator into the loss:

$\vec\rho(x)$6

equivalently an interpolation from regularly sampled values. The network is optimized through $\vec\rho(x)$7 rather than the raw field $\vec\rho(x)$8, so the learned solution is the projection onto the subspace reconstructible from uniform samples under the chosen kernel and spacing. The cutoff is tied to the grid by the Nyquist condition $\vec\rho(x)$9, and the construction is compatible with MLPs, hash grids, and hybrid fields (Shabanov et al., 2024).

BandRC shifts band limitation to the activation function. Its hidden-layer activation is the impulse response of a shifted raised-cosine filter with bandwidth parameter $[-\pi,\pi]^d$0, fixed roll-off $[-\pi,\pi]^d$1, and band shift $[-\pi,\pi]^d$2. In frequency space,

$[-\pi,\pi]^d$3

so the passband is centered at a nonzero frequency when $[-\pi,\pi]^d$4. The parameters $[-\pi,\pi]^d$5 are produced per layer by a harmonizer network fed by task-specific deep priors, while per-layer normalization to the unit circle is used for stability. This design is intended to mitigate spectral bias and improve sensitivity to both local and global structure (Thennakoon et al., 16 May 2025).

TOBACO uses an MFN-inspired coordinate network to represent a compositionally graded alloy design field. The sinusoidal input layers are

$[-\pi,\pi]^d$6

and the hidden recurrence is

$[-\pi,\pi]^d$7

The total bandwidth is the sum of the per-layer bandwidths, and the method sets

$[-\pi,\pi]^d$8

so that the represented composition field respects a prescribed maximum spatial gradation. Unlike many bounded-output INRs, TOBACO does not use output activations such as sigmoids, because the bounds and partition of unity are enforced through constraints and log barriers rather than by nonlinear output squashing (Chandrasekhar et al., 14 Aug 2025).

4. Multiscale representation, anti-aliasing, and reconstruction behavior

Band-limited coordinate networks are often motivated by multiscale reconstruction. BACON attaches readout heads at intermediate depths so that early outputs are low-pass and later outputs add higher-bandwidth refinements. A distinctive claim is that this multiresolution decomposition can be learned even from single-scale supervision: the intermediate outputs become band-limited approximations consistent with their cumulative cutoffs, rather than merely unsupervised by-products. Because the frequency support is bounded by design, downsampling can be anti-aliased by selecting the output whose bandwidth matches the target sampling grid, and upsampling does not create frequencies above the designed cutoff. Over finite intervals, the same sinusoidal construction also implies periodic extrapolation outside the training domain (Lindell et al., 2021).

BANF addresses a related problem from a different angle. It constructs a frequency decomposition through successive low-pass fits to residuals, effectively producing a Laplacian-style decomposition directly in neural-field space. The coarse representation is explicitly filtered before any downstream grid sampling, which is particularly important for marching cubes and other lattice-based operations. In the reported experiments, this simple sampling-aware modification produces measurable gains at coarse levels: on DIV2K $[-\pi,\pi]^d$9 images, BANF reaches PSNR $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$0 versus BACON $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$1 and PNF $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$2; for 3D SDF fitting on Asian Dragon at $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$3, Chamfer-L2 $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$4 improves from $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$5 for iNGP to $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$6 for BANF; and in inverse rendering on Synthetic NeRF at $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$7, Chamfer Distance $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$8 improves from $$c_k = \frac{1}{(2\pi)^d}\int_{x \in [-\pi,\pi]^d} f(x)e^{-j x^T k}\,dx,$$9 for NeUS to $K$0 for BANF (Shabanov et al., 2024).

BACON’s own empirical profile is different. In multiscale radiance fields, the reported average PSNR across scales is approximately $K$1 dB for NeRF, $K$2 dB for Mip-NeRF, and $K$3 dB for BACON. The architecture is therefore not presented as uniformly dominant over all scale-aware baselines; rather, its distinctive feature is an analytically controllable spectrum, lower parameter requirements at coarse outputs, and interpretable scale selection. In SDF fitting, BACON’s highest-resolution outputs are reported as comparable to Fourier Features, SIREN, and Nglod baselines, while its low-resolution outputs remain smooth and free of the coarse angular artifacts attributed to ReLU-based multiresolution schemes (Lindell et al., 2021).

Activation-level band limitation has also been used for signal and geometric reconstruction. BandRC reports an average PSNR improvement of $K$4 dB over the nearest counterpart across a diverse image dataset, a $K$5 dB increase in PSNR for denoising, and a $K$6 dB improvement over the nearest state-of-the-art method for $K$7 super-resolution. On Kodak image 20 specifically, BandRC exceeds INCODE by $K$8 dB PSNR and WIRE and SIREN by $K$9 dB and $c_k=0$0 dB, respectively. In 3D occupancy reconstruction on a Thai statue represented on a $c_k=0$1 grid, it reports IoU values of $c_k=0$2 for ReLU+PE, $c_k=0$3 for SIREN, $c_k=0$4 for WIRE, $c_k=0$5 for INCODE, and $c_k=0$6 for BandRC (Thennakoon et al., 16 May 2025).

5. Manufacturability-constrained topology optimization

In topology optimization for compositionally graded alloys, a band-limited coordinate neural network is used not primarily for representation quality but for direct enforcement of manufacturing constraints. TOBACO represents the composition field as

$c_k=0$7

with the network weights $c_k=0$8 acting as the design variables. This yields a mesh-independent representation that can be queried at arbitrary coordinates, supports high-resolution post-optimization sampling, and is differentiated end-to-end using JAX automatic differentiation. The paper states that the sensitivity of the loss to network weights is obtained automatically as $c_k=0$9 (Chandrasekhar et al., 14 Aug 2025).

The key theoretical device is Bernstein’s inequality. For a real-valued, band-limited $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$00 with bandwidth $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$01 and $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$02,

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$03

For composition fractions satisfying $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$04 and a manufacturing gradation constraint $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$05, the implied cutoff is

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$06

TOBACO therefore enforces the maximum gradation constraint implicitly by setting the sinusoidal input frequencies so that the total network bandwidth equals the allowable cutoff, rather than by adding explicit local gradient inequalities to the optimization problem (Chandrasekhar et al., 14 Aug 2025).

The optimization problem couples this parametrization to thermo-elastic and thermal PDEs, mass and bound constraints, and a partition-of-unity condition. The constrained problem in the network weights is written as minimizing $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$07 subject to $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$08, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$09, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$10, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$11, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$12, and $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$13, then converted to an unconstrained log-barrier loss

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$14

The implementation uses Adam with learning rate $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$15, gradient clipping with norm threshold $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$16, a structured bilinear quad mesh with $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$17 elements on a $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$18 domain, JAX-FEM for GPU-accelerated PDE solves, and termination after a maximum of $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$19 iterations or when $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$20 (Chandrasekhar et al., 14 Aug 2025).

The reported results make the trade-off between spectral restriction and objective value explicit. In a 2D elastic compliance validation with bandlimit $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$21, the band-limited design attains compliance $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$22, while a standard density-based topology-optimization baseline without gradation limit attains $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$23 and exhibits higher frequencies in the FFT. In a thermo-elastic bandwidth ablation, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$24, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$25, and $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$26 yield compliance values $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$27, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$28, and $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$29, respectively, showing that larger allowable bandwidth expands the design space. For anisotropic limits, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$30 gives compliance $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$31 and $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$32 gives $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$33. In a 3D turbine-blade example with maximum mass $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$34, sweeping $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$35 produces relative compliances $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$36, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$37, and $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$38 (Chandrasekhar et al., 14 Aug 2025).

6. Expressivity limits, trade-offs, and recurrent misconceptions

Band limitation improves analyzability and sampling behavior, but it is not neutral with respect to expressivity. In the ridgelet analysis of shallow random networks, for a bounded parameter set $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$39 and self-admissible bounded activation $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$40, the best attainable $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$41 approximation error over band-limited parameter distributions satisfies

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$42

Under $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$43, this lower bound is the $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$44 energy of the ridgelet spectrum outside the accessible parameter set. The result applies regardless of whether the hidden parameters are random or deterministic, and it shows that bounded-support random features cannot be universal whenever the target has nonzero ridgelet energy in $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$45 (Li et al., 2020).

The same paper makes the rate dependence on bandwidth and target smoothness explicit. For $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$46, the remaining irreducible error decreases as $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$47 increases and tends to zero only in the limit $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$48. Empirically, with sigmoid activation in 1D regression, $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$49 and width $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$50 do not achieve small error for a target with $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$51, whereas $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$52 and $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$53 achieve excellent performance. For the mixed-frequency target

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$54

$f_\theta : \mathbb{R}^r \to \mathbb{R}^c$55 fits the low-frequency quadratic component but fails to capture the oscillatory cosine term, while $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$56 and $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$57 fit both modes well (Li et al., 2020).

A common misconception is that band-limited always means the output network function is exactly band-limited. The sampling-theoretic analysis of finite coordinate MLPs makes a subtler statement: the target $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$58 may be band-limited, while the network interpolant $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$59 is only required to have sufficiently fast spectral decay, such as $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$60 in one dimension. Another misconception is that all band-limited coordinate networks impose the same constraint. BACON constrains cumulative sinusoidal frequencies, BANF constrains the reconstruction operator used during training, BandRC constrains the activation frequency response and learns a band center, TOBACO constrains bandwidth to encode a gradient bound, and the ridgelet work constrains hidden-parameter support (Giryes, 2020, Lindell et al., 2021, Shabanov et al., 2024, Thennakoon et al., 16 May 2025, Chandrasekhar et al., 14 Aug 2025, Li et al., 2020).

Across these formulations, the principal trade-off is consistent. Band limitation supports anti-aliasing, predictable behavior at unsupervised points, multiscale reconstruction, smoother interpolation, or manufacturability, but it can suppress high-frequency detail or reduce the feasible design space. The literature states this directly in several places: TOBACO’s compliance increases from $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$61 to $f_\theta : \mathbb{R}^r \to \mathbb{R}^c$62 when a gradation-constrained band-limited design replaces an unconstrained baseline; BANF notes that linear interpolation gives an approximate low-pass with leakage; BACON’s periodic extrapolation can be undesirable outside the supervised domain; and BandRC requires numerical care near the removable singularities of the raised-cosine impulse response and uses per-layer normalization for stability (Chandrasekhar et al., 14 Aug 2025, Shabanov et al., 2024, Lindell et al., 2021, Thennakoon et al., 16 May 2025).

A plausible synthesis is that band-limited coordinate neural networks are best understood not as a single architecture class but as a spectral design principle. In that sense, the field spans exact architectural band-limits, effective low-pass training operators, activation-engineered passbands, sampling-theoretic guarantees for interpolation, and application-specific frequency constraints. What unifies these strands is the replacement of opaque coordinate MLP behavior with explicit statements about which frequencies are representable, suppressed, or guaranteed under a given parametrization.