arxiv: 2605.11452 · v1 · submitted 2026-05-12 · 🧮 math.NA · cs.NA

Recognition: no theorem link

Resolving the Gibbs Phenomenon in Fractional Fourier Series via Inverse Polynomial Reconstruction

Faiza Afzal , Xu Xiao

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Pith reviewed 2026-05-13 02:20 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords fractional Fourier seriesGibbs phenomenoninverse polynomial reconstructionGegenbauer polynomialsspectral methodsnumerical convergencepiecewise analytic functions

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The pith

The inverse polynomial reconstruction method extends to fractional Fourier series and removes the Gibbs phenomenon independently of the rotation angle.

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

This paper shows that the inverse polynomial reconstruction method can be adapted to fractional Fourier series by deriving a transformation matrix that maps Gegenbauer polynomial coefficients to the observed fractional Fourier coefficients. The method reconstructs the underlying function directly from the spectral data rather than from the corrupted partial sum, which eliminates the oscillatory artifacts that appear near jumps in both real and imaginary parts. An L infinity error bound proves exponential convergence whenever the function is analytic, and the matrix conditioning turns out to depend only on the Gegenbauer parameter and polynomial degree, not on the fractional angle. A reader would care because fractional Fourier transforms are used in signal processing and optics where ringing near discontinuities can corrupt interpretations, and the new approach removes that artifact uniformly across angles.

Core claim

The paper claims that extending the inverse polynomial reconstruction method to fractional Fourier series produces a fractional transformation matrix whose conditioning is governed by an alpha-independent Gram matrix; this yields an L infinity error estimate that guarantees exponential convergence for analytic functions and, in numerical tests on piecewise analytic functions, completely eliminates the Gibbs phenomenon.

What carries the argument

The inverse polynomial reconstruction method, which enforces that the fractional Fourier coefficients of a Gegenbauer polynomial expansion exactly match the given spectral data.

Load-bearing premise

The target functions must be piecewise analytic or at least smooth away from jumps, and the Gegenbauer parameter must be chosen so the Gram matrix stays well-conditioned at the chosen polynomial degree.

What would settle it

Persistent oscillations near discontinuities in the reconstructed series for a simple piecewise analytic test function such as a step, even after the polynomial degree is increased, would show that the Gibbs phenomenon has not been eliminated.

Figures

Figures reproduced from arXiv: 2605.11452 by Faiza Afzal, Xu Xiao.

**Figure 1.** Figure 1: Conditioning of the coefficient matrix Wα for several values of the Gegenbauer parameter λ. (a) The condition number κ(Wα) as a function of the truncation level m. (b)–(c) The largest and smallest singular values σmax(Wα) and σmin(Wα), separating the two mechanisms that drive the growth of κ(Wα). (d) The condition number as a function of the transform angle α, illustrating its α-independence. the classical… view at source ↗

**Figure 2.** Figure 2: Reconstruction of six test functions with [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Figure 3: L ∞ error decay of the fractional IPRM for six test functions with α = π/4 and λ = 0.75. The horizontal axis is the Gegenbauer truncation degree m. All functions exhibit exponential convergence, with rates determined by the size of the analyticity region ρ. reconstruction accuracy should be insensitive to the transform angle α, since the underlying Gram matrix Gr is independent of α (Proposition 2). To ver… view at source ↗

**Figure 4.** Figure 4: Insensitivity of the reconstruction error to the transform angle [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗

read the original abstract

The fractional Fourier series generalizes the classical Fourier series by introducing a rotation angle $\alpha$ in the time-frequency plane, but inherits the Gibbs phenomenon for piecewise smooth functions. Unlike the classical setting, the chirp modulation factor renders the fractional partial sum complex-valued, corrupting both real and imaginary components simultaneously and making direct adaptation of classical remedies insufficient. The Inverse Polynomial Reconstruction Method (IPRM) resolves the Gibbs phenomenon by enforcing that the Fourier coefficients of a Gegenbauer polynomial expansion match the given spectral data, rather than projecting the corrupted partial sum onto a polynomial basis. This paper extends the IPRM to fractional Fourier series for the first time. The fractional transformation matrix is derived and its conditioning is shown to be governed by an $\alpha$-independent Gram matrix, which reveals the dependence on the Gegenbauer parameter $\lambda$ and the polynomial degree $m$, while being entirely insensitive to the transform angle. An $L^{\infty}$ error estimate is established, guaranteeing exponential convergence for analytic functions. Numerical experiments on piecewise analytic test functions demonstrate complete elimination of the Gibbs phenomenon and confirm the theoretical predictions.

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.

Desk Editor's Note private letter to a colleague

This extends IPRM to fractional Fourier series with a clean alpha-independent Gram matrix result, but the missing explicit bounds on conditioning for large m is the main gap.

read the letter

The punchline is that this is the first time IPRM has been adapted to fractional Fourier series, and the authors derive a transformation matrix whose conditioning depends only on the Gegenbauer parameters, not on the rotation angle alpha. That independence is a genuine simplification over what one might expect from the chirp modulation in the fractional case. They also give an L-infinity error bound that recovers the usual exponential convergence for analytic functions and show numerically that the Gibbs ringing disappears on piecewise analytic examples. Those are the concrete advances. The numerics are presented as confirming the theory, which is the right way to close the loop. The soft spot is exactly the one the stress test flags. The abstract says the dependence on lambda and m is revealed, yet there is no growth estimate for the condition number of the Gram matrix and no practical rule for picking lambda as m increases. If cond(G) grows faster than the convergence rate can offset, the linear solve for the Gegenbauer coefficients will lose accuracy before the error bound becomes relevant. I would have liked to see either a theorem bounding the condition number or at least a table of computed condition numbers versus m for a few fixed lambda values. Without that, the stability claim rests on the assumption that a good lambda can always be chosen, which is plausible but not yet demonstrated for the regime where one would actually use high m. This paper is aimed at numerical analysts and signal-processing people who already work with fractional Fourier transforms and need a Gibbs fix. A reader who knows the classical IPRM literature will see the extension immediately and can check the matrix derivation themselves. It is worth sending to a serious referee because the core construction is new, the claims are falsifiable, and the gap on conditioning is fixable rather than fatal. I would recommend review with a request to add either bounds or systematic numerical conditioning data in revision.

Referee Report

2 major / 2 minor

Summary. The paper extends the Inverse Polynomial Reconstruction Method (IPRM) to fractional Fourier series in order to resolve the Gibbs phenomenon for piecewise smooth functions. It derives the fractional transformation matrix, proves that its conditioning is controlled by an α-independent Gram matrix G(λ,m) arising from Gegenbauer inner products, establishes an L^∞ error estimate that guarantees exponential convergence for analytic functions, and reports numerical experiments on piecewise analytic test functions that demonstrate complete elimination of Gibbs ringing.

Significance. If the conditioning analysis and error bounds are rigorous, the work supplies a stable, angle-independent reconstruction procedure that generalizes classical IPRM to the fractional Fourier setting. The α-independence of the Gram matrix is a genuine technical strength, as it decouples reconstruction stability from the rotation angle. The exponential convergence claim for analytic functions, if supported by a complete proof, would be a useful theoretical guarantee.

major comments (2)

[Abstract (and the derivation of the fractional transformation matrix)] The central stability claim rests on the Gram matrix G(λ,m) remaining well-conditioned for the chosen m. The abstract asserts that the dependence on λ and m is revealed, yet no explicit growth bound on cond(G(λ,m)) or concrete selection rule λ(m) is supplied that would keep the condition number from outpacing the exponential convergence rate. This is load-bearing for the numerical reliability of the linear solve that recovers the Gegenbauer coefficients.
[Theoretical Analysis (error estimate section)] The L^∞ error estimate is stated to guarantee exponential convergence for analytic functions. Without a proof sketch, the precise dependence of the constant on cond(G) and on the distance to the nearest singularity, it is impossible to confirm that the bound remains valid once the reconstruction step is included.

minor comments (2)

[Numerical Experiments] The abstract refers to numerical experiments that demonstrate complete Gibbs elimination and confirm theoretical predictions, but supplies no quantitative error tables, convergence rates, or comparisons with direct fractional Fourier summation.
[Introduction / Preliminaries] Define the precise form of the fractional Fourier series and the entries of the transformation matrix at the first appearance rather than deferring all notation to later sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive evaluation of the work's significance, and the constructive comments on stability and error analysis. We address each major comment below.

read point-by-point responses

Referee: [Abstract (and the derivation of the fractional transformation matrix)] The central stability claim rests on the Gram matrix G(λ,m) remaining well-conditioned for the chosen m. The abstract asserts that the dependence on λ and m is revealed, yet no explicit growth bound on cond(G(λ,m)) or concrete selection rule λ(m) is supplied that would keep the condition number from outpacing the exponential convergence rate. This is load-bearing for the numerical reliability of the linear solve that recovers the Gegenbauer coefficients.

Authors: We agree that an explicit a priori bound on cond(G(λ,m)) and a concrete selection rule for λ(m) would strengthen the stability analysis. The manuscript derives the closed-form expression for the α-independent Gram matrix G(λ,m) arising from the Gegenbauer inner products and shows how its entries depend on λ and m; this already reveals the dependence. Numerical results in the experiments section confirm that the chosen parameters keep the linear solve stable and yield accurate reconstructions. In the revision we will add a remark (or short appendix) providing a growth estimate for cond(G) based on standard bounds for Gegenbauer polynomials and propose a practical rule λ(m) ∼ m^β with β small enough that cond(G) grows at most polynomially, which remains compatible with exponential convergence. revision: partial
Referee: [Theoretical Analysis (error estimate section)] The L^∞ error estimate is stated to guarantee exponential convergence for analytic functions. Without a proof sketch, the precise dependence of the constant on cond(G) and on the distance to the nearest singularity, it is impossible to confirm that the bound remains valid once the reconstruction step is included.

Authors: The L^∞ estimate combines the classical exponential approximation rate of Gegenbauer polynomials (controlled by the distance to the nearest singularity in the complex plane) with the operator norm of the reconstruction step, which is bounded by cond(G). The manuscript states the final bound but does not spell out the intermediate steps that track these two factors separately. We will insert a concise proof sketch in the revised theoretical section that explicitly factors the constant into C(analyticity radius) · cond(G) and verifies that the exponential rate is preserved whenever cond(G) grows slower than the approximation factor, which is ensured by the parameter choices already used in the numerics. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation rests on independent Gegenbauer properties and analytic approximation theory

full rationale

The paper derives the fractional transformation matrix explicitly and shows its conditioning is controlled by the standard α-independent Gram matrix arising from Gegenbauer inner products. This is a direct algebraic result, not a self-definition or renaming. The L∞ error bound follows from classical polynomial approximation rates for analytic functions and does not presuppose the numerical stability of the solve as part of its statement. No load-bearing step reduces to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors. The extension of IPRM is presented as a methodological application whose theoretical guarantees are independent of the fractional case outputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The approach relies on standard properties of Gegenbauer polynomials and the definition of the fractional Fourier transform; no new entities are postulated.

free parameters (2)

Gegenbauer parameter λ
Controls the weight in the polynomial basis and affects matrix conditioning; chosen by the user.
polynomial degree m
Degree of the reconstruction polynomial; trades off accuracy and conditioning.

axioms (2)

standard math Gegenbauer polynomials form an orthogonal basis with respect to the weight (1-x²)^λ on [-1,1]
Invoked when matching coefficients to spectral data.
domain assumption The fractional Fourier transform is well-defined and invertible for the function class considered
Required for the partial sums and coefficient matching to make sense.

pith-pipeline@v0.9.0 · 5487 in / 1315 out tokens · 37567 ms · 2026-05-13T02:20:30.042178+00:00 · methodology

discussion (0)

Reference graph

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