Distributional Point Values for Borel and Symmetric Borel Derivatives

Subhasis Ray

arxiv: 2606.12472 · v1 · pith:7Y4NSCGPnew · submitted 2026-06-10 · 🧮 math.GM

Distributional Point Values for Borel and Symmetric Borel Derivatives

Subhasis Ray This is my paper

Pith reviewed 2026-06-27 07:56 UTC · model grok-4.3

classification 🧮 math.GM

keywords Borel derivativesymmetric Borel derivativedistributional point valueŁojasiewicz point valueregular distributiongeneralized derivativeone-sided derivative

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The pith

Finite first and second symmetric Borel derivatives imply symmetric distributional point values of T_f' and T_f'' respectively.

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

The paper connects Borel derivatives, defined via local averages of difference quotients, to distributional point values in the Łojasiewicz sense. It proves that finite symmetric Borel derivatives of order one or two produce the corresponding symmetric distributional point values for the first and second derivatives of the regular distribution T_f generated by f. A reader would care because this supplies a concrete way to obtain distributional local behavior from more elementary average-based derivatives. Additional results address one-sided Borel derivatives and the role of Borel smoothness in lifting symmetric values to full point values.

Core claim

Writing T_f for the regular distribution generated by f, we prove that finite first and second symmetric Borel derivatives give symmetric distributional point values of T_f' and T_f'', respectively. For the first symmetric derivative, Borel smoothness is used as a sufficient condition to pass from the symmetric point value to the full Łojasiewicz point value. We also prove that the one-sided Borel derivatives determine the right and left distributional point values of T_f', and that the ordinary Borel derivative gives the full point value when the two one-sided averages agree.

What carries the argument

Symmetric Borel derivatives, defined through local averages of difference quotients, which establish the symmetric distributional point values of T_f' and T_f'' in the Łojasiewicz sense.

If this is right

The one-sided Borel derivatives determine the right and left distributional point values of T_f'.
When the two one-sided averages agree, the ordinary Borel derivative yields the full distributional point value.
Borel smoothness suffices to obtain the full Łojasiewicz point value from the symmetric point value in the first-order case.
Examples show that the second-order symmetric implication does not extend automatically to stronger statements.

Where Pith is reading between the lines

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

The link may let analysts verify distributional local properties by computing averages rather than testing against arbitrary test functions.
Higher-order extensions could follow if suitable smoothness or agreement conditions are added.
The approach could support numerical checks of distributional behavior where local difference quotients are easier to evaluate.

Load-bearing premise

Symmetric distributional point values are defined exactly in the classical Łojasiewicz sense and its symmetric variants, and the relevant Borel derivatives are finite at the point.

What would settle it

Direct computation, for a concrete function with finite second symmetric Borel derivative, of whether the symmetric distributional limit of T_f'' with test functions equals the Borel value.

read the original abstract

Borel and symmetric Borel derivatives are generalized derivatives defined through local averages of difference quotients. Distributional point values, in the sense of {\L}ojasiewicz and its symmetric variants, are a classical way of describing the local value of a distribution. This paper connects these two ideas. Writing $T_f$ for the regular distribution generated by $f$, we prove that finite first and second symmetric Borel derivatives give symmetric distributional point values of $T_f'$ and $T_f''$, respectively. For the first symmetric derivative, Borel smoothness is used as a sufficient condition to pass from the symmetric point value to the full {\L}ojasiewicz point value. We also prove that the one-sided Borel derivatives determine the right and left distributional point values of $T_f'$, and that the ordinary Borel derivative gives the full point value when the two one-sided averages agree. Examples show why the second-order symmetric result cannot be strengthened automatically.

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 proves that finite symmetric Borel derivatives imply symmetric distributional point values for the first and second derivatives of T_f, with one-sided versions and explicit limits on strengthening.

read the letter

The core result here is that finite first and second symmetric Borel derivatives of f yield symmetric distributional point values for T_f' and T_f'' in the Łojasiewicz sense. The paper also shows one-sided Borel derivatives control the corresponding one-sided distributional values, and that the ordinary Borel derivative gives the full point value when the one-sided averages match. Borel smoothness is used to upgrade the symmetric case to the full point value in order one.

The work does a clean job of stating the direct implications from the definitions and backing them with theorems. The examples that block automatic strengthening at order two are a useful addition; they keep the claims honest rather than overstated. The argument structure looks non-circular and stays within the classical setup for these objects.

The main limitation is narrowness. This is a technical link between two existing notions in real analysis, and the abstract gives no sign it opens broader questions or simplifies existing proofs elsewhere. Without the full proofs in front of me I cannot check the estimates in detail, but nothing in the stated claims suggests a load-bearing gap. The dependence on finite derivatives and the classical definition of the point values is explicit and reasonable.

This is for specialists in distributional derivatives or generalized differentiation. A reader already working in that corner will get the precise statements and the counterexamples for free. It is not foundational, but the claims are specific enough and the approach careful enough that it deserves a serious referee who knows the literature on Borel and Łojasiewicz point values.

Referee Report

0 major / 3 minor

Summary. The paper proves that finite first and second symmetric Borel derivatives of a function f yield symmetric distributional point values (in the Łojasiewicz sense) for the first and second derivatives of the regular distribution T_f. It further shows that Borel smoothness suffices to upgrade the symmetric point value to the full Łojasiewicz point value in the first-order case, that one-sided Borel derivatives determine the corresponding one-sided distributional point values of T_f', and that the ordinary Borel derivative yields the full point value when the one-sided averages agree. Explicit examples demonstrate why the second-order symmetric result cannot be strengthened in general.

Significance. If the stated implications hold, the results provide a precise bridge between Borel-type generalized derivatives (defined via local averages of difference quotients) and classical distributional point values. This link may be useful in real analysis when working with functions of limited smoothness whose distributional derivatives are under study. The paper's inclusion of sharpness examples for the second-order case is a strength, as it clarifies the precise scope of the theorems rather than overclaiming.

minor comments (3)

The abstract refers to 'Borel smoothness' as a sufficient condition for upgrading to the full Łojasiewicz point value but does not define the term or provide a reference; a short inline definition or citation in the introduction would improve readability.
Notation for the symmetric distributional point value and the one-sided variants should be introduced with explicit formulas (or clear references to Łojasiewicz's original definitions) at the first use to avoid ambiguity for readers unfamiliar with the symmetric variants.
The examples limiting the second-order result are mentioned in the abstract; ensuring they are presented with explicit function definitions and computed derivatives in a dedicated section would strengthen the manuscript's clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes direct implications between finite symmetric Borel derivatives and symmetric distributional point values (in the Łojasiewicz sense) for the associated derivatives of the regular distribution T_f. These are presented as consequences of the respective definitions, with Borel smoothness invoked only as a sufficient condition for strengthening the first-order case and explicit examples limiting the second-order case. No self-citations, fitted parameters renamed as predictions, ansatzes smuggled via prior work, or uniqueness theorems are load-bearing in the derivation chain. The argument remains self-contained against the classical definitions without reducing any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard mathematical axioms and definitions from distribution theory and generalized differentiation, with no free parameters or invented entities introduced.

axioms (1)

standard math Standard definitions and properties of Borel derivatives and Łojasiewicz distributional point values
The paper connects existing concepts using their classical definitions.

pith-pipeline@v0.9.1-grok · 5680 in / 1112 out tokens · 24795 ms · 2026-06-27T07:56:53.314677+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 10 canonical work pages

[1]

Khintchine, Recherches sur la structure des fonctions mesurables,Fundam

A. Khintchine, Recherches sur la structure des fonctions mesurables,Fundam. Math.9(1) (1927), 212–279. https://doi.org/10.4064/fm-9-1-212-279

work page doi:10.4064/fm-9-1-212-279 1927
[2]

Marcinkiewicz and A

J. Marcinkiewicz and A. Zygmund, On the differentiability of functions and summability of trigonometrical series,Fundam. Math.26(1) (1936), 1–43. https://doi.org/10.4064/fm-26-1-1- 43

work page doi:10.4064/fm-26-1-1- 1936
[3]

Sargent, The Borel derivates of a function,Proc

W.L.C. Sargent, The Borel derivates of a function,Proc. Lond. Math. Soc.s2–38(1) (1935), 180–196. https://doi.org/10.1112/plms/s2-38.1.180

work page doi:10.1112/plms/s2-38.1.180 1935
[4]

Mukhopadhyay,Higher Order Derivatives, Chapman and Hall/CRC, Boca Raton, 2012

S.N. Mukhopadhyay,Higher Order Derivatives, Chapman and Hall/CRC, Boca Raton, 2012. https://doi.org/10.1201/b11617

work page doi:10.1201/b11617 2012
[5]

Ray and A

S. Ray and A. Garai, On Borel derivative,Bull. Calcutta Math. Soc.106(4) (2014), 273–280

2014
[6]

Łojasiewicz, Sur la valeur et la limite d’une distribution en un point,Stud

S. Łojasiewicz, Sur la valeur et la limite d’une distribution en un point,Stud. Math.16(1) (1957), 1–36. https://doi.org/10.4064/sm-16-1-1-36

work page doi:10.4064/sm-16-1-1-36 1957
[7]

Phase retrieval from Gabor measurements

J. Vindas Diaz and R. Estrada, Distributional point values and convergence of Fourier series and integrals,J. Fourier Anal. Appl.13(5) (2007), 551–576. https://doi.org/10.1007/s00041- 006-6015-z. 11

work page doi:10.1007/s00041- 2007
[8]

Vindas Diaz and R

J. Vindas Diaz and R. Estrada, Distributionally regulated functions,Stud. Math.181(3) (2007), 211–236. https://doi.org/10.4064/sm181-3-2

work page doi:10.4064/sm181-3-2 2007
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J.VindasDiazandR.Estrada, Onthejumpbehaviorofdistributionsandlogarithmicaverages, J. Math. Anal. Appl.347(2) (2008), 597–606. https://doi.org/10.1016/j.jmaa.2008.06.014

work page doi:10.1016/j.jmaa.2008.06.014 2008
[10]

Vindas Diaz and R

J. Vindas Diaz and R. Estrada, On the order of summability of the Fourier inversion formula, Anal. Theory Appl.26(1) (2010), 13–42. https://doi.org/10.1007/s10496-010-0013-3

work page doi:10.1007/s10496-010-0013-3 2010
[11]

Estrada and K

R. Estrada and K. Kellinsky-Gonzalez, Distributional point values and delta sequences,Indag. Math. (N.S.)33(4) (2022), 719–735. https://doi.org/10.1016/j.indag.2022.02.002. 12

work page doi:10.1016/j.indag.2022.02.002 2022

[1] [1]

Khintchine, Recherches sur la structure des fonctions mesurables,Fundam

A. Khintchine, Recherches sur la structure des fonctions mesurables,Fundam. Math.9(1) (1927), 212–279. https://doi.org/10.4064/fm-9-1-212-279

work page doi:10.4064/fm-9-1-212-279 1927

[2] [2]

Marcinkiewicz and A

J. Marcinkiewicz and A. Zygmund, On the differentiability of functions and summability of trigonometrical series,Fundam. Math.26(1) (1936), 1–43. https://doi.org/10.4064/fm-26-1-1- 43

work page doi:10.4064/fm-26-1-1- 1936

[3] [3]

Sargent, The Borel derivates of a function,Proc

W.L.C. Sargent, The Borel derivates of a function,Proc. Lond. Math. Soc.s2–38(1) (1935), 180–196. https://doi.org/10.1112/plms/s2-38.1.180

work page doi:10.1112/plms/s2-38.1.180 1935

[4] [4]

Mukhopadhyay,Higher Order Derivatives, Chapman and Hall/CRC, Boca Raton, 2012

S.N. Mukhopadhyay,Higher Order Derivatives, Chapman and Hall/CRC, Boca Raton, 2012. https://doi.org/10.1201/b11617

work page doi:10.1201/b11617 2012

[5] [5]

Ray and A

S. Ray and A. Garai, On Borel derivative,Bull. Calcutta Math. Soc.106(4) (2014), 273–280

2014

[6] [6]

Łojasiewicz, Sur la valeur et la limite d’une distribution en un point,Stud

S. Łojasiewicz, Sur la valeur et la limite d’une distribution en un point,Stud. Math.16(1) (1957), 1–36. https://doi.org/10.4064/sm-16-1-1-36

work page doi:10.4064/sm-16-1-1-36 1957

[7] [7]

Phase retrieval from Gabor measurements

J. Vindas Diaz and R. Estrada, Distributional point values and convergence of Fourier series and integrals,J. Fourier Anal. Appl.13(5) (2007), 551–576. https://doi.org/10.1007/s00041- 006-6015-z. 11

work page doi:10.1007/s00041- 2007

[8] [8]

Vindas Diaz and R

J. Vindas Diaz and R. Estrada, Distributionally regulated functions,Stud. Math.181(3) (2007), 211–236. https://doi.org/10.4064/sm181-3-2

work page doi:10.4064/sm181-3-2 2007

[9] [9]

J.VindasDiazandR.Estrada, Onthejumpbehaviorofdistributionsandlogarithmicaverages, J. Math. Anal. Appl.347(2) (2008), 597–606. https://doi.org/10.1016/j.jmaa.2008.06.014

work page doi:10.1016/j.jmaa.2008.06.014 2008

[10] [10]

Vindas Diaz and R

J. Vindas Diaz and R. Estrada, On the order of summability of the Fourier inversion formula, Anal. Theory Appl.26(1) (2010), 13–42. https://doi.org/10.1007/s10496-010-0013-3

work page doi:10.1007/s10496-010-0013-3 2010

[11] [11]

Estrada and K

R. Estrada and K. Kellinsky-Gonzalez, Distributional point values and delta sequences,Indag. Math. (N.S.)33(4) (2022), 719–735. https://doi.org/10.1016/j.indag.2022.02.002. 12

work page doi:10.1016/j.indag.2022.02.002 2022