Fractal Based Rational Cubic Trigonometric Zipper Interpolation with Positivity Constraints

A. K. Sharma; K. R. Tyada

arxiv: 2509.06532 · v3 · submitted 2025-09-08 · 🧮 math.NA · cs.NA

Fractal Based Rational Cubic Trigonometric Zipper Interpolation with Positivity Constraints

A. K. Sharma , K. R. Tyada This is my paper

Pith reviewed 2026-05-18 18:25 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords fractal interpolationpositivity preservationrational cubic trigonometriczipper fractalshape parametersscaling factorsconvergence analysisnumerical interpolation

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The pith

A new zipper fractal interpolation method preserves positivity in datasets by constraining shape parameters and scaling factors.

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

The paper introduces Rational Cubic Trigonometric Zipper Fractal Interpolation Functions (RCTZFIFs) that combine rational cubic trigonometric basis functions with a zipper fractal structure. The central goal is to interpolate positive data points while ensuring the resulting curve never goes negative. The authors derive explicit bounds on scaling factors and shape parameters, then prove that choices within those bounds enforce positivity. They also establish convergence of the fractal interpolants to the data-generating function through error bounds. Numerical examples illustrate cases where the new method stays positive but a standard reference interpolant crosses below zero.

Core claim

By carefully selecting the signature, shape parameters, and scaling factors within derived bounds, the RCTZFIFs effectively preserve the positive nature of the data, as compared to a reference interpolant that may violate this property.

What carries the argument

Rational Cubic Trigonometric Zipper Fractal Interpolation Functions (RCTZFIFs) built from rational cubic trigonometric pieces inside a zipper fractal iteration, with free shape parameters and scaling factors whose bounds are chosen to enforce positivity.

If this is right

The RCTZFIFs converge to the underlying classical fractal functions and then to the data-generating function under the stated error bounds.
Positivity is maintained for any positive input data when parameters lie inside the derived constraints.
The method supplies greater local shape control than non-fractal cubic trigonometric interpolants while still respecting positivity.
Numerical visualisations confirm that the fractal version avoids the negativity seen in the reference interpolant on the same data.

Where Pith is reading between the lines

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

The same bounding technique on scaling factors might be adapted to preserve monotonicity or convexity in future zipper fractal schemes.
The approach could be tested on time-series data from fields such as population growth or asset prices where negativity is meaningless.
Local variation of the zipper signature might allow the positivity constraints to be tightened only in regions where the data approaches zero.

Load-bearing premise

The derived constraints on scaling factors and shape parameters are sufficient to guarantee positivity preservation for arbitrary positive datasets without introducing new violations or overly restricting the fractal flexibility.

What would settle it

Take any set of strictly positive data points, choose signature, shape parameters, and scaling factors strictly inside the paper's derived bounds, and check whether the resulting RCTZFIF ever becomes negative; if it does for any such choice, the preservation claim is false.

Figures

Figures reproduced from arXiv: 2509.06532 by A. K. Sharma, K. R. Tyada.

**Figure 1.** Figure 1: Visualisations of RCTZFIFs under different parameter configurations. Figure 1a illustrates the RCTZFIF without positivity constraints imposed on shape parameters and scaling factor. Nevertheless, values as mentioned in [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

read the original abstract

We propose a novel fractal based interpolation scheme termed Rational Cubic Trigonometric Zipper Fractal Interpolation Functions (RCTZFIFs) designed to model and preserve the inherent geometric property, positivity, in given datasets. The method employs a combination of rational cubic trigonometric functions within a zipper fractal framework, offering enhanced flexibility through shape parameters and scaling factors. Rigorous error analysis is presented to establish the convergence of the proposed zipper fractal interpolants to the underlying classical fractal functions, and subsequently, to the data-generating function. We derive necessary constraints on the scaling factors and shape parameters to ensure positivity preservation. By carefully selecting the signature, shape parameters, and scaling factors within these bounds, we construct a class of RCTZFIFs that effectively preserve the positive nature of the data, as compared to a reference interpolant that may violate this property. Numerical experiments and visualisations demonstrate the efficacy and robustness of our approach in preserving positivity while offering fractal flexibility.

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 paper builds a positivity-preserving rational cubic trigonometric zipper fractal interpolant with derived parameter bounds and error analysis, though the global guarantee under iteration looks like the part to check closely.

read the letter

The main thing here is a new scheme called RCTZFIFs that uses rational cubic trigonometric functions inside a zipper fractal setup and adds constraints on scaling factors and shape parameters to keep the result non-negative for positive data. They also give an error analysis showing convergence to the classical non-fractal case and to the underlying function, plus some numerical examples that compare favorably to a reference interpolant that goes negative. The fusion of the trigonometric rational cubics with the zipper structure is the concrete new piece, and the direct derivation of the bounds from the positivity requirement avoids obvious circularity. The visuals and tests seem to illustrate the flexibility without losing the shape property. The soft spot is whether those bounds actually deliver global positivity once the fractal iterations run with nonzero scaling. Local non-negativity at knots and first-level bounds do not automatically extend to the full attractor, since repeated compositions can still produce sign changes in between. If the proof only handles the initial maps or the fixed-point equation without checking the iterated behavior across all subintervals, the constraints could turn out too loose for arbitrary positive datasets. That is the part worth a referee's attention rather than a fatal flaw. This is aimed at people already working on fractal interpolation and shape-preserving approximation in numerical analysis or geometric modeling. A reader who needs flexible positive interpolants with some fractal roughness would find the construction and examples useful. The work shows enough technical engagement with the literature and derivations that it deserves a serious referee to verify the positivity proof and the tightness of the bounds.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Rational Cubic Trigonometric Zipper Fractal Interpolation Functions (RCTZFIFs) constructed from rational cubic trigonometric basis functions within a zipper fractal interpolation scheme. The authors derive constraints on scaling factors, shape parameters, and signature to enforce positivity preservation for positive datasets, present an error analysis establishing convergence of the zipper fractal interpolants to the underlying classical functions and data-generating function, and provide numerical experiments showing that suitably chosen parameters yield positive interpolants where a reference method may violate positivity.

Significance. If the derived constraints are shown to be sufficient and non-vacuous for the fractal attractor, the work would extend shape-preserving interpolation techniques by combining trigonometric rational splines with zipper fractals, offering additional flexibility via free parameters while maintaining positivity. The explicit error analysis and parameter bounds, if rigorously verified, represent a constructive contribution to fractal approximation methods applicable to positive data modeling in visualization and scientific computing.

major comments (2)

[Positivity constraints derivation] The derivation of bounds on scaling factors and shape parameters (abstract and positivity section) is presented as ensuring the fixed point remains non-negative, yet the argument relies on local non-negativity at knots and basis-function bounds; it does not explicitly demonstrate that these bounds prevent sign changes under iterative composition of the zipper IFS when scaling factors are nonzero for arbitrary positive data.
[Error analysis] The error analysis establishes convergence to the classical (non-fractal) reference interpolant, but does not address whether the positivity constraints survive the fractal perturbation uniformly; a uniform bound or counterexample test for cases where the reference is positive yet the attractor violates positivity would be required to support the central claim.

minor comments (2)

[Numerical experiments] The numerical experiments section would be strengthened by tabulating the chosen scaling factors and shape parameters alongside the resulting maximum and minimum values of the interpolant to verify that the derived bounds are tight and non-vacuous.
[Preliminaries / Zipper framework] Clarify the precise definition and role of the 'signature' parameter in the zipper construction, as its interaction with the scaling factors is central to the positivity argument but is introduced without explicit notation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each major comment in detail below, proposing revisions where appropriate to enhance the rigor of our arguments.

read point-by-point responses

Referee: The derivation of bounds on scaling factors and shape parameters (abstract and positivity section) is presented as ensuring the fixed point remains non-negative, yet the argument relies on local non-negativity at knots and basis-function bounds; it does not explicitly demonstrate that these bounds prevent sign changes under iterative composition of the zipper IFS when scaling factors are nonzero for arbitrary positive data.

Authors: We appreciate the referee pointing out the need for a more explicit treatment of the iterative process in the positivity preservation proof. The constraints on the scaling factors and shape parameters are derived such that each application of the zipper IFS operator maps non-negative functions to non-negative functions when the data is positive. Specifically, the rational cubic trigonometric basis functions are non-negative, and the bounds ensure that the weighted combinations remain positive. To make this rigorous for the attractor, we will revise the positivity section to include an inductive argument: assuming the k-th iterate is non-negative, the (k+1)-th iterate is shown to be non-negative under the given bounds, for any nonzero scaling factors satisfying the inequalities. This will explicitly demonstrate that sign changes are prevented throughout the iteration process for arbitrary positive data. revision: yes
Referee: The error analysis establishes convergence to the classical (non-fractal) reference interpolant, but does not address whether the positivity constraints survive the fractal perturbation uniformly; a uniform bound or counterexample test for cases where the reference is positive yet the attractor violates positivity would be required to support the central claim.

Authors: The error analysis demonstrates that the RCTZFIF converges to the classical rational cubic trigonometric interpolant as the scaling factors tend to zero. The positivity constraints are formulated to hold for the fractal case independently, ensuring the attractor preserves positivity even when the classical interpolant does not. We acknowledge that an explicit uniform bound on the difference in positivity (or a counterexample search) would further support this. We will add numerical experiments in the revised manuscript that test cases where the reference interpolant is positive, applying the constraints to the fractal version and verifying no violation occurs, along with a remark discussing the uniform convergence in the context of positivity preservation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; positivity constraints derived as explicit design goal

full rationale

The paper derives bounds on scaling factors and shape parameters directly from the mathematical requirement that the RCTZFIF remain non-negative on the interval, which is the stated objective rather than a hidden reduction of the result to its inputs. Error analysis for convergence to the classical interpolant is presented separately and does not rely on the positivity result. No self-citations, fitted parameters renamed as predictions, or uniqueness theorems imported from prior author work appear as load-bearing steps in the provided derivation chain. The construction is self-contained against the external benchmark of preserving positivity for given positive data.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The construction rests on standard properties of fractal interpolation and trigonometric functions; the main additions are derived bounds rather than new free parameters or entities.

free parameters (2)

shape parameters
Control local curve shape and are constrained to enforce positivity.
scaling factors
Govern fractal perturbation strength and are bounded to maintain positivity.

axioms (2)

standard math Zipper fractal interpolants converge to classical fractal functions under suitable conditions on scaling factors.
Invoked when the abstract states rigorous error analysis establishing convergence.
domain assumption Rational cubic trigonometric functions satisfy standard positivity and interpolation properties when parameters lie in certain ranges.
Background assumption for the trigonometric basis used in the scheme.

pith-pipeline@v0.9.0 · 5692 in / 1357 out tokens · 47141 ms · 2026-05-18T18:25:59.673537+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

?

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

We derive necessary constraints on the scaling factors and shape parameters to ensure positivity preservation... 0 ≤ λ_j < min{ a_i, f_{j+ε_j}/f_1, f_{j+1-ε_j}/f_n }
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

RCTZFIFs... zipper fractal framework... signature ε = (ε1,...,ε_{n-1}) ∈ {0,1}^{n-1}

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.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

A., Ali, J

Abbas, M., Majid, A. A., Ali, J. Md.: Positivity-preserving C2 rational cubic spline inter- polation, ScienceAsia, 39, 208-213, 2013. 11

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[2]

Z., and Hussain, M.: Shape-preserving curve interpolation

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I.: On shape preserving quadratic spline interpolation

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Navascués, M. A., Chand, A. K. B., V eedu, V . P ., and Sebastián, M. V .: Fractal interpola- tion functions: a short survey . Applied Mathematics, 5(12), 1834-1841, 2014

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F.: Fractal functions and interpolation

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F., Elton, J., Hardin, D., and Massopust, P

Barnsley, M. F., Elton, J., Hardin, D., and Massopust, P . : Hidden V ariable Fractal Interpo- lation Functions, SIAM J. Math. Anal., 20(5), 1218–1242, 1989

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A.: Fractal polynomial interpolation , Zeitschrift für Analysis und ihre Anwendungen, 24(2), 401–418, 2005

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A., Ali, J

Abbas, M., Majid, A. A., Ali, J. Md.: Monotonicity-preserving C2 rational cubic spline for monotone data , Applied Mathematics and Computation, 219, 2885-2895, 201 2

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Chand, A. K. B., Tyada, K. R.: Constrained shape preserving rational cubic fractal inter- polation functions, Rocky Mt. J. Math., 48 (1), 75–105, 2018

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Chand, A. K. B., Tyada, K. R.: Shape preserving constrained and monotonic rational quintic fractal interpolation functions , Int. J. Adv. Eng. Sci. Appl. Math., 10 (1), 15–33, 2018

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[14]

Chand, A. K. B., Tyada, K. R.: Constrained 2D data interpolation using rational cubic fractal functions, in: Mathematical Analysis and its Applications, in: Sprin ger Proc. Math. Stat., 143, Springer, New Delhi, 593–607, 2015

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[15]

Chand, A. K. B., and Tyada, K. R.: Positivity preserving rational cubic trigonometric fractal interpolation functions, Mathematics and Computing, 139, 187, 2015

work page 2015
[16]

R., Chand, A

Tyada, K. R., Chand, A. K. B., and Sajid, M.: Shape preserving rational cubic trigono- metric fractal interpolation functions . Mathematics and Computers in Simulation, 190, 866-891

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Aseev, V . V . E., Tetenov, A. V ., and Kravchenko, A. S.: On selfsimilar Jordan curves on the plane. Siberian Mathematical Journal, 44(3), 379-386, 2003

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Chand, A. K. B., Vijender, N., Viswanathan, P ., and Tete nov, A. V .: Afﬁne zipper fractal interpolation functions BIT Numerical Mathematics, 60(2), 319-344, 2020. 12

work page 2020
[19]

Chand, A. K. B.: Zipper fractal functions with variable scalings . Advances in the Theory of Nonlinear Analysis and its Application, 6(4), 481-501, 2 022

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Jha, S., and Chand, A. K. B.: Zipper rational quadratic fractal interpolation function s. In Proceedings of the Fifth International Conference on Mathe matics and Computing: ICMC 2019 (pp. 229-241). Singapore: Springer Singapore, 2020. 13

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[1] [1]

A., Ali, J

Abbas, M., Majid, A. A., Ali, J. Md.: Positivity-preserving C2 rational cubic spline inter- polation, ScienceAsia, 39, 208-213, 2013. 11

work page 2013

[2] [2]

Z., and Hussain, M.: Shape-preserving curve interpolation

Sarfraz, M., Hussain, M. Z., and Hussain, M.: Shape-preserving curve interpolation. Inter- national Journal of Computer Mathematics, 89(1), 35-53, 20 12

work page

[3] [3]

I.: On shape preserving quadratic spline interpolation

Schumaker, L. I.: On shape preserving quadratic spline interpolation . SIAM Journal on Numerical Analysis, 20(4), 854-864, 1983

work page 1983

[4] [4]

Liu, Z., and Liu, S.: C1 shape-preserving rational quadratic/linear interpolati on splines with necessary and sufﬁcient conditions , Symmetry, 17(6), 815, 2025

work page 2025

[5] [5]

A., Chand, A

Navascués, M. A., Chand, A. K. B., V eedu, V . P ., and Sebastián, M. V .: Fractal interpola- tion functions: a short survey . Applied Mathematics, 5(12), 1834-1841, 2014

work page 2014

[6] [6]

F.: Fractal functions and interpolation

Barnsley, M. F.: Fractal functions and interpolation . Constructive Approximation, 2(4), 303–329, 1986

work page 1986

[7] [7]

F., and Harrington, A

Barnsley, M. F., and Harrington, A. N.: The calculus of fractal interpolation functions . Journal of Approximation Theory, 57(1), 14-34, 1989

work page 1989

[8] [8]

F., Elton, J., Hardin, D., and Massopust, P

Barnsley, M. F., Elton, J., Hardin, D., and Massopust, P . : Hidden V ariable Fractal Interpo- lation Functions, SIAM J. Math. Anal., 20(5), 1218–1242, 1989

work page 1989

[9] [9]

Chand, A. K. B., and Navascués, M. A.: Generalized Hermite fractal interpolation , Rev Real Academia de Ciencias, Zaragoza, vol. 64, 107–120, 2009

work page 2009

[10] [10]

A.: Fractal polynomial interpolation , Zeitschrift für Analysis und ihre Anwendungen, 24(2), 401–418, 2005

Navascués, M. A.: Fractal polynomial interpolation , Zeitschrift für Analysis und ihre Anwendungen, 24(2), 401–418, 2005

work page 2005

[11] [11]

A., Ali, J

Abbas, M., Majid, A. A., Ali, J. Md.: Monotonicity-preserving C2 rational cubic spline for monotone data , Applied Mathematics and Computation, 219, 2885-2895, 201 2

work page

[12] [12]

Chand, A. K. B., Tyada, K. R.: Constrained shape preserving rational cubic fractal inter- polation functions, Rocky Mt. J. Math., 48 (1), 75–105, 2018

work page 2018

[13] [13]

Chand, A. K. B., Tyada, K. R.: Shape preserving constrained and monotonic rational quintic fractal interpolation functions , Int. J. Adv. Eng. Sci. Appl. Math., 10 (1), 15–33, 2018

work page 2018

[14] [14]

Chand, A. K. B., Tyada, K. R.: Constrained 2D data interpolation using rational cubic fractal functions, in: Mathematical Analysis and its Applications, in: Sprin ger Proc. Math. Stat., 143, Springer, New Delhi, 593–607, 2015

work page 2015

[15] [15]

Chand, A. K. B., and Tyada, K. R.: Positivity preserving rational cubic trigonometric fractal interpolation functions, Mathematics and Computing, 139, 187, 2015

work page 2015

[16] [16]

R., Chand, A

Tyada, K. R., Chand, A. K. B., and Sajid, M.: Shape preserving rational cubic trigono- metric fractal interpolation functions . Mathematics and Computers in Simulation, 190, 866-891

work page

[17] [17]

Aseev, V . V . E., Tetenov, A. V ., and Kravchenko, A. S.: On selfsimilar Jordan curves on the plane. Siberian Mathematical Journal, 44(3), 379-386, 2003

work page 2003

[18] [18]

Chand, A. K. B., Vijender, N., Viswanathan, P ., and Tete nov, A. V .: Afﬁne zipper fractal interpolation functions BIT Numerical Mathematics, 60(2), 319-344, 2020. 12

work page 2020

[19] [19]

Chand, A. K. B.: Zipper fractal functions with variable scalings . Advances in the Theory of Nonlinear Analysis and its Application, 6(4), 481-501, 2 022

work page

[20] [20]

Jha, S., and Chand, A. K. B.: Zipper rational quadratic fractal interpolation function s. In Proceedings of the Fifth International Conference on Mathe matics and Computing: ICMC 2019 (pp. 229-241). Singapore: Springer Singapore, 2020. 13

work page 2019