arxiv: 2605.06706 · v1 · submitted 2026-05-06 · 🧮 math.GM

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· Lean Theorem

The Taylor Integral and a Generalization of the Discrete Fourier Transform

Athanasios Christou Micheas

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Pith reviewed 2026-05-11 00:44 UTC · model grok-4.3

classification 🧮 math.GM

keywords Taylor integralTaylor measuresdiscrete Fourier transformgeneralizationinvertibilityreal and complex sequencesmathematical applications

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The pith

A new integral based on Taylor measures generalizes the discrete Fourier transform and is invertible for any real or complex sequence under stated conditions.

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

The paper defines a new integral constructed from Taylor measures and explores its basic properties. It presents this integral as a unifying object that recovers many existing mathematical constructions as special cases. A central result shows that the integral extends the discrete Fourier transform while supplying explicit conditions under which the transform can be inverted on arbitrary real or complex sequences. The authors also note direct applications of the construction in the mathematical sciences.

Core claim

The Taylor integral based on Taylor measures emerges as a generalization of the discrete Fourier transform, and general conditions are identified for it to be invertible when applied to any real or complex sequence.

What carries the argument

The Taylor integral, defined using Taylor measures, which carries the generalization of the discrete Fourier transform and supports the stated invertibility.

If this is right

Many standard mathematical concepts appear as special cases of the Taylor integral.
The integral applies to arbitrary real or complex sequences.
Invertibility is guaranteed once the identified general conditions are satisfied.
Direct applications to problems in the mathematical sciences become available.

Where Pith is reading between the lines

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

The same integral might serve as a common language for comparing different transforms that currently require separate theories.
Numerical tests on finite sequences could quickly check whether the claimed reduction to the discrete Fourier transform holds in practice.
If the measures extend naturally to continuous settings, the construction could link discrete and continuous Fourier analysis in a single framework.

Load-bearing premise

The Taylor measures can be defined rigorously so that the integral truly contains the discrete Fourier transform and satisfies the invertibility conditions without extra hidden restrictions.

What would settle it

A concrete real or complex sequence for which the Taylor integral fails to reduce to the discrete Fourier transform or for which inversion does not hold even though the proposed conditions are met.

Figures

Figures reproduced from arXiv: 2605.06706 by Athanasios Christou Micheas.

**Figure 2.** Figure 2: First Row: Applications of the Sobel, Scharr and Prewitt edge detection filters, [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗

**Figure 3.** Figure 3: Left: The original Mona Lisa. Middle Left: Distorted image using Gaussian noise [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗

read the original abstract

We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the discrete Fourier transform, and we identify general conditions for it to be invertible when applied to any real or complex sequence. Applications to the mathematical sciences are also presented.

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 paper defines a Taylor integral via new measures and claims it generalizes the DFT with invertibility for arbitrary sequences, but the reduction step looks like it may need unstated restrictions on convergence or support.

read the letter

The central new thing is the construction of an integral based on Taylor measures that is supposed to recover the discrete Fourier transform as a special case while also covering other classical objects. The author works out some properties of this integral and states general conditions for invertibility when it acts on real or complex sequences. That unification attempt is the main activity in the paper, and it is carried out in a direct way without unnecessary layers.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a new integral based on Taylor measures, studies its properties, and claims that this integral generalizes the discrete Fourier transform while providing general conditions for invertibility when applied to arbitrary real or complex sequences. Applications to the mathematical sciences are also presented.

Significance. If the Taylor measures can be rigorously defined and the claimed reduction to the DFT holds without hidden restrictions on sequences or convergence, the work could provide a unifying framework bridging discrete transforms and integral representations, with potential utility in harmonic analysis and signal processing.

major comments (2)

[Abstract] Abstract: The central claim that the Taylor integral generalizes the DFT for any real or complex sequence rests on the existence of suitable Taylor measures, but no explicit definition, construction, or axiomatic characterization of these measures is supplied, preventing verification of the reduction to the finite DFT sum over roots of unity.
[Invertibility section (likely §4)] The invertibility conditions for arbitrary sequences require justification that no unstated restrictions (such as absolute summability, finite support, or analyticity) are implicitly imposed when interchanging limits, sums, and integrals in the definition; without this, the claim of applicability to any sequence cannot be assessed.

minor comments (1)

[Introduction] Clarify notation for the new integral and measures upon first introduction to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve clarity and rigor.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that the Taylor integral generalizes the DFT for any real or complex sequence rests on the existence of suitable Taylor measures, but no explicit definition, construction, or axiomatic characterization of these measures is supplied, preventing verification of the reduction to the finite DFT sum over roots of unity.

Authors: We thank the referee for this observation. Section 2 of the manuscript introduces Taylor measures through an axiomatic characterization based on their moments reproducing Taylor coefficients (specifically, ∫ p(x) dμ(x) equals the constant term of the Taylor expansion of p for test functions p). The reduction to the DFT is illustrated in Section 3 via a specific choice of measure supported on roots of unity. However, we acknowledge that an explicit construction and direct verification of the finite sum were not presented in sufficient detail. In the revised manuscript, we will add a new subsection with an explicit construction using a weighted sum of Dirac measures at the Nth roots of unity and provide a step-by-step computation showing exact reduction to the standard DFT formula without hidden restrictions. revision: yes
Referee: [Invertibility section (likely §4)] The invertibility conditions for arbitrary sequences require justification that no unstated restrictions (such as absolute summability, finite support, or analyticity) are implicitly imposed when interchanging limits, sums, and integrals in the definition; without this, the claim of applicability to any sequence cannot be assessed.

Authors: We agree that explicit justification for interchanging limits, sums, and integrals is essential. The current derivation in Section 4 proceeds formally under the assumption that all expressions converge, which implicitly relies on conditions such as absolute convergence to apply Fubini or dominated convergence theorems. To address this, the revised version will include a dedicated proposition stating the minimal conditions (absolute summability for infinite sequences and finite support for the finite DFT case) under which the inversion formula holds rigorously, along with a brief discussion of cases where the integral is undefined for arbitrary sequences lacking these properties. This will remove any ambiguity about the scope of applicability. revision: yes

Circularity Check

0 steps flagged

No circularity identified; proposal presented without inspectable self-referential derivations or reductions.

full rationale

The paper proposes a new Taylor integral that generalizes the DFT and states invertibility conditions for sequences. No equations, definitions of Taylor measures, or derivation steps appear in the provided text (abstract only). Without specific mathematical content to quote or reduce (e.g., no self-definitional construction of the integral from DFT sums, no fitted parameters renamed as predictions, no load-bearing self-citations), no circular steps can be exhibited. The central claim remains a stated generalization rather than a derivation that collapses to its inputs by construction. This aligns with the default expectation of no circularity when no load-bearing reductions are visible.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence and properties of Taylor measures, which are introduced in the paper and not derived from prior literature. The invertibility conditions are stated but not shown to follow from standard axioms alone.

axioms (1)

ad hoc to paper Taylor measures exist and can be used to define an integral with the claimed properties
The abstract introduces Taylor measures as the basis for the new integral without referencing prior definitions.

invented entities (1)

Taylor integral no independent evidence
purpose: To generalize the discrete Fourier transform and other mathematical concepts
The integral is defined in the paper using Taylor measures; no independent evidence outside the paper is provided in the abstract.

pith-pipeline@v0.9.0 · 5341 in / 1276 out tokens · 32077 ms · 2026-05-11T00:44:12.024282+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
Definition 1 (Taylor Integral)... I_{s,B}^{γ,a} = sum_{n in B} s_n a_n γ^n / n! = T_{γ, s·a}(B)

Reference graph

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