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arxiv: 2605.03549 · v1 · submitted 2026-05-05 · 🧮 math.NA · cs.NA

Fourier Residual Networks Achieve Spectral Accuracy for Discontinuous Functions

Owen Davis , Mohammad Motamed , Olof Runborg This is my paper

Pith reviewed 2026-05-07 14:37 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Fourier residual networksspectral convergencediscontinuous functionsHermite interpolationfixed-point iterationtrigonometric polynomialsneural network approximationnumerical analysis
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The pith

Fourier residual networks achieve spectral accuracy for discontinuous functions without requiring periodicity or continuity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs Fourier residual networks that approximate one-dimensional functions with jumps in the function value or its derivatives, as well as smooth functions, at spectral rates. Spectral accuracy means the error drops exponentially with network depth rather than polynomially. This removes two standard restrictions of Fourier series: the target function must be periodic and continuous. It also avoids the Barron-space limitation common in other neural approximation results. The construction uses classical fixed-point iteration realized through residual layers that perform trigonometric Hermite interpolation, and the claims are checked with both exact constructions and randomized numerical tests.

Core claim

Fourier residual networks achieve spectral convergence for piecewise continuous functions that may have jump discontinuities and for fully smooth functions. The networks realize a fixed-point iteration that employs Hermite interpolation by trigonometric polynomials, and this realization works uniformly without assuming periodicity or continuity of the target function and without restricting the function class to Barron spaces.

What carries the argument

Residual network layers that implement fixed-point iteration via Hermite interpolation with trigonometric polynomials.

If this is right

  • Spectral convergence holds uniformly across functions with jumps in the function or its derivatives.
  • The same networks attain spectral rates for fully smooth functions without change in architecture.
  • No periodicity assumption is needed, unlike classical linear Fourier approximation.
  • The result applies outside Barron-type function spaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same residual construction could be tested on time-dependent problems whose solutions develop discontinuities.
  • Higher-dimensional versions would require extending the trigonometric interpolation step while preserving the fixed-point structure.
  • Randomized training variants mentioned in the experiments may inherit the same convergence guarantees if the iteration is preserved.

Load-bearing premise

The fixed-point iteration combined with Hermite interpolation by trigonometric polynomials can be realized exactly as the architecture of a Fourier residual network for the broad class of discontinuous and smooth functions considered.

What would settle it

Compute the approximation error for a step function or other discontinuous test case as network depth increases; if the error fails to decay exponentially with depth, the spectral-convergence claim is false.

Figures

Figures reproduced from arXiv: 2605.03549 by Mohammad Motamed, Olof Runborg, Owen Davis.

Figure 1
Figure 1. Figure 1: Schematic of the Fourier network fL, which employs a residual-style architecture. The first layer consists of W neurons and takes x as input, producing the output f1(x) = g1(x). Each subsequent layer ℓ ≥ 2 has two parallel branches, each with W neurons: an upper branch that computes gℓ(x) from x, and a lower branch that computes hℓ(fℓ−1(x)) from fℓ−1(x), which is the output of the previous layer. The outpu… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the Fourier network FFN used to approximate f. The first L layers generate SL(x); a parallel branch at layer L computes sin(x), and their sum yields ZL(x). The final layer consists of two branches: the upper branch computes RW (x) from x using W + 2(m + 1) neurons, while the lower branch computes H(ZL(x)) from ZL(x) using 2(m + 1) neurons. These two outputs are added to ZL(x) to produce the fi… view at source ↗
Figure 3
Figure 3. Figure 3: Approximation of the sign function on [−1, 1] using the Fourier ResNet (solid blue) and the truncated Fourier series (dashed red) for L = 5 (left) and L = 20 (right). The exact sign function is shown as a thin solid black line for reference. While the truncated Fourier series exhibits Gibbs oscillations, the ResNet approximation remains monotonic and fully resolves the Gibbs phenomenon. 0 5 10 15 L 10-5 10… view at source ↗
Figure 4
Figure 4. Figure 4: L 1 approximation error of the Fourier ResNet (solid) and the truncated Fourier series (dashed) for the sign function, plotted against the number of terms L on a log-linear scale. The Fourier ResNet achieves exponential convergence, while the truncated Fourier series converges only algebraically. 20 view at source ↗
Figure 5
Figure 5. Figure 5: Approximation of a piecewise smooth function using a truncated Fourier series (red) and a view at source ↗
Figure 6
Figure 6. Figure 6: Decay in the spatial support of spurious oscillations in the Fourier ResNet approximation of the view at source ↗
Figure 7
Figure 7. Figure 7: L 2 error of the Fourier ResNet approximation of the piecewise smooth function (29) versus the width parameter W = 10 · 2 j for j = 0, . . . , 6, shown on a log-log scale. The depth is fixed at L = 20 to isolate the second term in the error bound (25). Curves correspond to m ∈ {1, 2, 3, 4}. The results confirm spectral convergence of Fourier ResNet, with empirical rates close to O(W −m−3/2 ), while the tru… view at source ↗
Figure 8
Figure 8. Figure 8: L 2 approximation error of the Fourier ResNet applied to the hat function (30), plotted against the width parameter W = 10 · 2 j for j = 0, . . . , 6, on a log-log scale. The depth is fixed at L = 20, isolating the effect of W on the second term of the error bound (25). Curves for m ∈ {1, 2, 3, 4} confirm spectral convergence of Fourier ResNet, with empirical rates consistent with O(W −m−3/2 ), while the t… view at source ↗
Figure 9
Figure 9. Figure 9: MSE in shallow Fourier ResNet approximations (solid circle) as a function of width parameter view at source ↗
Figure 10
Figure 10. Figure 10: On the left, the true target function (black solid) along with approximations from a Fourier view at source ↗
Figure 11
Figure 11. Figure 11: On the left, predictions on the test data from a Fourier ResNet of fixed width view at source ↗
read the original abstract

We present a constructive approximation framework for analyzing the expressive power of Fourier residual networks in approximating a broad class of one-dimensional functions. Our study covers both piecewise continuous functions -- including those with jump discontinuities in the function and its derivatives -- and fully smooth functions. We show that Fourier residual networks achieve spectral convergence without requiring periodicity or continuity, thereby overcoming key limitations of classical linear Fourier approximation and nonlinear methods, without being restricted to Barron-type function spaces. Our approach builds on classical techniques from approximation theory, including fixed-point iteration and Hermite interpolation by trigonometric polynomials. We support our theoretical results with numerical experiments based on both the constructed approximations and a randomized algorithm developed in our earlier work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript presents a constructive approximation framework for Fourier residual networks approximating one-dimensional piecewise-continuous functions (including jumps in value or derivatives) and smooth functions. It claims that these networks achieve spectral convergence without requiring periodicity or continuity, via fixed-point iteration combined with Hermite interpolation by trigonometric polynomials, and supports the theory with numerical experiments on both constructed approximations and a randomized algorithm from prior work.

Significance. If the central construction is rigorously verified, the result would be significant for approximation theory and neural network expressivity: it offers a parameter-free, constructive route to spectral accuracy for discontinuous functions that classical linear Fourier methods cannot handle and that is not limited to Barron-type spaces. The explicit use of classical tools (fixed-point iteration, Hermite trig interpolation) and the provision of reproducible numerical support are positive features.

major comments (3)
  1. [Theoretical Framework] The claim that the fixed-point iteration using Hermite interpolation by trigonometric polynomials can be exactly realized as a Fourier residual network (identity plus Fourier layer) while preserving spectral rates for jump-discontinuous functions is load-bearing; the abstract asserts this but the explicit embedding, contractivity proof in a uniform norm, and error bounds must be shown in detail (see Theoretical Framework section).
  2. [Theoretical Results] Trigonometric polynomials are globally periodic and C^∞; the manuscript must demonstrate how the residual construction cancels these constraints for functions with unknown jump locations without degrading the spectral rate or requiring a priori discontinuity information (this directly addresses the uniformity claim over the stated function class).
  3. [Numerical Experiments] The numerical experiments (both constructed and randomized) should report explicit convergence rates (e.g., log-error vs. degree) for representative discontinuous test functions and compare against classical Fourier truncation and other residual architectures to confirm the claimed spectral behavior.
minor comments (2)
  1. [Notation] Clarify the precise definition of the Fourier residual layer and the iteration operator in the notation section to avoid ambiguity when reading the construction.
  2. [Introduction] Add a short discussion of related work on Hermite interpolation for discontinuous functions and on residual networks for spectral approximation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The comments identify key areas where additional detail and clarification will strengthen the presentation. We respond to each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: The claim that the fixed-point iteration using Hermite interpolation by trigonometric polynomials can be exactly realized as a Fourier residual network (identity plus Fourier layer) while preserving spectral rates for jump-discontinuous functions is load-bearing; the abstract asserts this but the explicit embedding, contractivity proof in a uniform norm, and error bounds must be shown in detail (see Theoretical Framework section).

    Authors: We agree that the explicit embedding of the fixed-point iteration into the Fourier residual network architecture, along with the contractivity argument in the uniform norm and the resulting error bounds, requires a more detailed exposition to make the load-bearing claim fully rigorous. In the revised manuscript we will expand the Theoretical Framework section with a self-contained derivation showing how each iteration step maps to an identity-plus-Fourier-layer residual block, the contraction mapping property in the uniform norm, and the spectral error estimates that hold for functions with jump discontinuities. revision: yes

  2. Referee: Trigonometric polynomials are globally periodic and C^∞; the manuscript must demonstrate how the residual construction cancels these constraints for functions with unknown jump locations without degrading the spectral rate or requiring a priori discontinuity information (this directly addresses the uniformity claim over the stated function class).

    Authors: The residual construction works by iteratively adding correction terms obtained from Hermite trigonometric interpolation; because the iteration converges to the target function in the uniform norm irrespective of periodicity, the accumulated residuals effectively remove the artificial periodicity and smoothness imposed by each trigonometric polynomial. The interpolation nodes are chosen globally and do not require prior knowledge of jump locations; the fixed-point iteration itself adapts to the locations of discontinuities. We will add a dedicated paragraph in the Theoretical Results section that spells out this cancellation mechanism and proves that the spectral rate is preserved uniformly over the stated function class. revision: partial

  3. Referee: The numerical experiments (both constructed and randomized) should report explicit convergence rates (e.g., log-error vs. degree) for representative discontinuous test functions and compare against classical Fourier truncation and other residual architectures to confirm the claimed spectral behavior.

    Authors: We concur that explicit quantitative reporting of convergence rates and systematic comparisons will make the numerical evidence more compelling. In the revised version we will augment the Numerical Experiments section with log-error versus degree plots for several representative discontinuous test functions, together with direct comparisons against classical Fourier truncation and alternative residual architectures. These additions will be placed alongside the existing constructed and randomized experiments. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for experimental support only; core derivation independent of self-references.

full rationale

The paper's central derivation constructs Fourier residual networks from classical fixed-point iteration combined with Hermite interpolation by trigonometric polynomials, which are standard tools in approximation theory and independent of the present work. Spectral convergence for discontinuous functions is claimed to follow from this realization without periodicity or continuity requirements. The sole self-reference is to a randomized algorithm from the authors' earlier work, used exclusively to generate additional numerical experiments that support (but do not define) the theoretical claims. No load-bearing step reduces the main result to a fitted parameter, self-definition, or unverified self-citation chain; the framework remains externally grounded in classical results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on classical approximation-theory tools (fixed-point iteration, Hermite interpolation by trig polynomials) whose details are not expanded in the abstract; no free parameters, new entities, or ad-hoc axioms are explicitly introduced.

axioms (2)
  • domain assumption Fixed-point iteration converges for the operator defining the residual network approximation
    Invoked to construct the network for both continuous and discontinuous targets
  • standard math Hermite interpolation by trigonometric polynomials exists and is stable for the target function class
    Used to achieve spectral rates without periodicity

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