Exact Deconvolution for Schwartz Kernels: From Polynomial Automorphisms to Recursive Inversion in Tempered Distributions

Alfredo Gonz\'alez-Calvin

arxiv: 2602.23780 · v2 · submitted 2026-02-27 · 📡 eess.SP · math.FA

Exact Deconvolution for Schwartz Kernels: From Polynomial Automorphisms to Recursive Inversion in Tempered Distributions

Alfredo Gonz\'alez-Calvin This is my paper

Pith reviewed 2026-05-15 19:08 UTC · model grok-4.3

classification 📡 eess.SP math.FA

keywords deconvolutionSchwartz kernelrecursive convolutiontempered distributionsWeierstrass transformexact inversionpolynomial automorphismsignal recovery

0 comments

The pith

Convolution with an even Schwartz kernel can be exactly inverted by taking the limit of finite recursive convolution sums that work on polynomials.

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

The paper establishes an exact deconvolution procedure that uses only repeated forward convolutions. It first shows that an even Schwartz kernel acts as an automorphism on the finite-degree polynomials, so the inverse on that space is a finite linear combination of repeated convolutions. The same finite-sum expression is then passed to the limit in the topologies of L1, L2, the Schwartz space, and tempered distributions, recovering the original element exactly. This construction supplies a new, explicitly computable iterative formula for the inverse Weierstrass transform that avoids classical ill-posed steps.

Core claim

The central claim is that the algebraic inverse for the automorphism induced by an even Schwartz kernel on polynomials extends by continuity: the finite-sum formula of recursive convolutions converges, in the respective topology, to the exact preimage of any element in L1, L2, the Schwartz space, or the space of tempered distributions.

What carries the argument

The limit of the finite linear combination of repeated convolutions that inverts the polynomial automorphism induced by the even Schwartz kernel.

If this is right

Any function or tempered distribution convolved with the kernel is recovered exactly by the limiting procedure.
The Weierstrass transform admits an explicit iterative numerical inverse that uses only forward convolutions.
The method supplies a mathematically exact and numerically stable algorithm for computational signal recovery.
Traditional regularization or truncation steps for ill-posed deconvolution are bypassed.

Where Pith is reading between the lines

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

The same limiting construction might apply to other kernels that induce automorphisms on polynomials.
Multi-dimensional versions or extensions to other spaces of distributions could be obtained by repeating the same passage to the limit.
The approach suggests that many integral transforms may possess exact iterative inverses once a dense algebraic sub-space inverse is known.

Load-bearing premise

The sequence of linear combinations of recursive convolutions converges to the original element in the topology of L1, L2, Schwartz space, or tempered distributions.

What would settle it

A concrete Schwartz kernel together with a function in L2 for which the limit of the recursive-convolution sums fails to recover the original function in the L2 norm.

read the original abstract

In this work, we construct an explicit, theoretically rigorous deconvolution method that relies entirely on iterative forward convolutions, thus can be numerically implemented. We first prove that convolution with an even Schwartz kernel acts as an automorphism on the vector space of finite-degree polynomials. Exploiting the parity of the kernel, we derive an exact algebraic inverse for this space, expressed uniquely as a finite linear combination of repeated convolutions. The core contribution of this work extends this algebraic inversion to infinite-dimensional function spaces, including $L^1(\mathbb{R})$, $L^2(\mathbb{R})$, the Schwartz space $\mathscr{S}(\mathbb{R})$, and the space of tempered distributions $\mathscr{S}'(\mathbb{R})$. By passing the finite-sum polynomial inversion formula to the limit, we demonstrate that an arbitrary function or distribution convolved with a Schwartz kernel can be exactly recovered in its respective topology. The resulting inverse is an explicitly computable limit of a sequence of linear combinations of recursive convolutions. As a primary application, this limit provides a fundamentally new, iterative numerical formula for the inverse of the Weierstrass Transform. By bypassing traditional numerically ill-posed inversion techniques, our method offers a mathematically exact and numerically robust algorithm for computational signal recovery.

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

The algebraic inverse on polynomials via finite repeated convolutions is concrete and looks new, but the claimed limit extension to L1, L2, Schwartz space and tempered distributions rests on an unshown mechanism.

read the letter

The paper's main contribution is an explicit algebraic inverse for convolution against an even Schwartz kernel when restricted to finite-degree polynomials, expressed as a finite linear combination of repeated forward convolutions. They then assert that passing this formula to a limit yields exact recovery for arbitrary elements of L1(R), L2(R), the Schwartz space and tempered distributions. The polynomial construction itself is the part that holds up on the abstract: the automorphism property follows from the kernel's smoothness and evenness, and the inverse formula is algebraic and exact on that finite-dimensional space. This avoids Fourier division or regularization and is directly implementable, which is useful for the Weierstrass-transform application they highlight. That piece deserves credit as a clean, reproducible algebraic result. The extension step is the soft spot. Non-zero polynomials are not in L1, L2 or Schwartz space because they fail to decay at infinity, so no sequence of them can converge in norm to a generic function in those spaces. The abstract gives no auxiliary device—cutoffs, mollification, or a weak-* limit in S'—to keep the approximants inside the target topology while the formula is applied. Without that, the continuous extension of the inverse operator is not demonstrated. The stress-test note identifies exactly this gap, and the abstract supplies no counter to it. A reader working on numerical deconvolution or distribution-theoretic inversion would find the polynomial case worth examining even if the general claim needs repair. The work shows clear thinking on the algebraic side and honest engagement with the numerical motivation, so it is coherent on its own terms. I would bring it to a reading group to walk through the limit construction in the full text. I would not cite it yet. It deserves peer review so referees can check whether the extension is properly justified or whether the polynomial result stands alone.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to develop an exact deconvolution technique for Schwartz kernels by proving that convolution with an even Schwartz kernel induces an automorphism on the vector space of finite-degree polynomials. It derives an algebraic inverse expressed as a finite linear combination of repeated convolutions and extends this to L^1(R), L^2(R), Schwartz space, and tempered distributions by taking the limit of the polynomial formula, yielding an iterative numerical method for inverting the Weierstrass transform.

Significance. If the limit passage is rigorously established, the result would provide a novel, parameter-free exact inversion method for a broad class of convolutions in signal processing and distribution theory. The explicit recursive form and avoidance of traditional ill-posed inverses represent a potentially important contribution, particularly for applications requiring stable numerical recovery of signals smoothed by Gaussian or similar kernels. The algebraic foundation on polynomials is a clear strength.

major comments (2)

[Abstract and core contribution paragraph] Abstract: the central extension claim states that 'by passing the finite-sum polynomial inversion formula to the limit' an arbitrary function or distribution can be recovered exactly in the respective topologies. However, since non-zero polynomials do not belong to L^1(R), L^2(R), or the Schwartz space (lacking decay at infinity), it is unclear how the sequence of approximants remains in the target space or converges in the respective topology. No auxiliary construction such as cutoffs, mollification, or weak-* convergence in S' is indicated to bridge this gap. This issue is load-bearing for the main result.
[Core contribution (extension to infinite-dimensional spaces)] The section describing the extension to infinite-dimensional spaces: the justification for the continuous extension of the inverse operator to L^1, L^2, S, and S' relies on an unverified limit argument whose mechanism is not supplied. A concrete proof or explicit construction showing convergence in the norm or weak topologies is required to support the claim of exact recovery.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments correctly identify that the extension from the polynomial automorphism to the infinite-dimensional spaces requires a more explicit construction and convergence proof than is currently supplied. We will revise the manuscript to address both points fully.

read point-by-point responses

Referee: [Abstract and core contribution paragraph] Abstract: the central extension claim states that 'by passing the finite-sum polynomial inversion formula to the limit' an arbitrary function or distribution can be recovered exactly in the respective topologies. However, since non-zero polynomials do not belong to L^1(R), L^2(R), or the Schwartz space (lacking decay at infinity), it is unclear how the sequence of approximants remains in the target space or converges in the respective topology. No auxiliary construction such as cutoffs, mollification, or weak-* convergence in S' is indicated to bridge this gap. This issue is load-bearing for the main result.

Authors: We agree that the abstract and the high-level statement of the extension leave the approximation mechanism underspecified. The manuscript currently asserts the limit passage without detailing how the polynomial approximants are projected into the target spaces. In the revision we will add an explicit auxiliary construction (compactly supported cutoffs followed by mollification) that produces a sequence lying in L^1, L^2, S, and S' respectively, together with a proof that the inverted sequence converges to the desired pre-image in the corresponding topology. The abstract will be updated to reflect this construction. revision: yes
Referee: [Core contribution (extension to infinite-dimensional spaces)] The section describing the extension to infinite-dimensional spaces: the justification for the continuous extension of the inverse operator to L^1, L^2, S, and S' relies on an unverified limit argument whose mechanism is not supplied. A concrete proof or explicit construction showing convergence in the norm or weak topologies is required to support the claim of exact recovery.

Authors: The referee correctly notes that the current text presents the limit argument at a summary level without a self-contained convergence proof. We will expand the relevant section with a complete argument: first define the approximating sequence via truncation and mollification so that each approximant is in the target space, then verify that the algebraic inverse applied to the approximants converges in the L^1, L^2, Schwartz, and weak-* S' topologies to the exact pre-image. This will be inserted as a new subsection immediately after the polynomial result. revision: yes

Circularity Check

0 steps flagged

No circularity: algebraic derivation on polynomials extended by limit without self-reference or fitted inputs

full rationale

The paper first establishes an automorphism property of even Schwartz kernels on the finite-dimensional vector space of polynomials (an algebraic fact independent of the target function spaces), derives an explicit finite-sum inverse formula consisting of repeated convolutions, and then states that the same formula is passed to a limit to obtain the inverse on L1, L2, S, and S'. This chain does not reduce any claimed result to a definition of itself, a fitted parameter renamed as a prediction, or a load-bearing self-citation. No ansatz is smuggled via prior work, and no known empirical pattern is merely renamed. The derivation is therefore self-contained against external algebraic and topological benchmarks; any potential gap in justifying the limit passage (e.g., density or convergence issues) is a question of correctness, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of Schwartz spaces and distributions together with the novel limit construction; no free parameters or new entities are introduced.

axioms (2)

domain assumption Convolution with an even Schwartz kernel is an automorphism on the space of finite-degree polynomials.
Invoked as the starting point for the algebraic inverse.
ad hoc to paper The finite-sum inversion formula extends to the limit in the topologies of L1, L2, Schwartz space, and tempered distributions.
This is the load-bearing step that carries the result to infinite dimensions.

pith-pipeline@v0.9.0 · 5521 in / 1179 out tokens · 27959 ms · 2026-05-15T19:08:37.652384+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

convolution with an even Schwartz kernel acts as an automorphism on the vector space of finite-degree polynomials... exact algebraic inverse... finite linear combination of repeated convolutions... passing the finite-sum polynomial inversion formula to the limit
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

lim n→∞ T^{-1}_{ε,n}(T_ε(f)) = f in L2, pointwise in L1, distributional sense in S'

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.