Shape Preserving Zipper Hidden Variable Fractal Interpolation Function

Chol Hui Yun; Kyong Ju Ri; Mi Gyong Ri; Yu Jong Pak

arxiv: 2604.09699 · v1 · submitted 2026-04-07 · 🧮 math.DS

Shape Preserving Zipper Hidden Variable Fractal Interpolation Function

Chol Hui Yun , Yu Jong Pak , Mi Gyong Ri , Kyong Ju Ri This is my paper

Pith reviewed 2026-05-10 19:43 UTC · model grok-4.3

classification 🧮 math.DS

keywords zipper fractal interpolationhidden variable IFSshape preservationboundednesspositivitypiecewise slopevertical scaling factorsiterated function systems

0 comments

The pith

Zipper hidden variable fractal interpolation functions preserve boundedness, positivity, and piecewise slope under conditions on vertical 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 a zipper hidden variable iterated function system to generate a new family of fractal interpolation functions with greater shape variety than standard ones. It derives explicit conditions on the vertical scaling factors that force these functions to respect the original data's bounds, stay positive when the data is positive, and keep the same local slopes between points. A reader would care because fractal interpolants often introduce unwanted wiggles or sign changes that break the intended shape, and this method supplies controllable parameters to avoid that while still allowing fractal roughness. The construction uses the extra freedom of hidden variables and zipper connections to achieve the preservation. Concrete examples confirm that the required scaling factors can be found and produce the desired functions.

Core claim

We introduce a ZHVIFS and construct univariate ZFIFs using the ZHVIFS. ZFIFs have more diverse shape than usual fractal interpolation functions, and the hidden variable iterated function system is more general than the iterated function system. Next, we find the conditions on vertical scaling factors for the ZFIFs to preserve boundedness, positivity and piecewise slope of the data set, which are demonstrated through examples.

What carries the argument

The zipper hidden variable iterated function system (ZHVIFS) that produces zipper fractal interpolation functions (ZFIFs) whose vertical scaling factors are tuned to enforce shape preservation.

If this is right

ZFIFs stay within the bounds of the given data when the scaling factors obey the stated inequalities.
Positive data sets produce strictly positive ZFIFs under the derived conditions on the scaling factors.
The piecewise slopes between consecutive data points are reproduced exactly by the ZFIF when the scaling factors meet the slope-preservation criteria.
The zipper and hidden-variable structure supplies extra parameters that allow shape preservation while still generating rough fractal curves.

Where Pith is reading between the lines

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

The same scaling-factor conditions might be adapted to enforce monotonicity or convexity preservation in addition to the three properties already treated.
Because hidden variables add degrees of freedom, the method could be applied to interpolation problems where standard fractal functions fail to match both shape and roughness requirements.
Numerical search algorithms for the scaling factors could be developed to automate the construction for large data sets.

Load-bearing premise

Suitable vertical scaling factors exist that are small enough to keep the ZHVIFS contractive while also satisfying the inequalities for boundedness, positivity, and piecewise slope preservation at the same time.

What would settle it

A concrete data set for which no choice of vertical scaling factors satisfies the contraction mapping requirement of the ZHVIFS together with all three preservation conditions simultaneously.

read the original abstract

In this paper, we study a new class of zipper fractal interpolation functions (ZFIFs) constructed using a zipper hidden variable iterated function system (ZHVIFS). ZFIFs have more diverse shape than usual fractal interpolation functions, and the hidden variable iterated function system is more general than the iterated function system. Firstly, we introduce a ZHVIFS and construct univariate ZFIFs using the ZHVIFS. Next, we find the conditions on vertical scaling factors for the ZFIFs to preserve boundedness, positivity and piecewise slop of the data set, which are demonstrated through examples.

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 builds ZFIFs from a zipper hidden-variable IFS and derives scaling-factor conditions to preserve positivity and piecewise slopes, but does not show those conditions are always solvable under contractivity.

read the letter

The main point is a construction of fractal interpolants that uses a zipper hidden variable iterated function system and then supplies inequalities on the vertical scaling factors so the attractor stays bounded, stays positive, and matches the slopes between data points. They define the ZHVIFS, show that its attractor is a continuous function passing through the given points, and work out the alpha bounds needed for each preservation property before checking them on a few data sets with plots. This adds the zipper mechanism on top of existing hidden-variable FIFs, which gives extra parameters for shape control while still staying inside the standard IFS fixed-point framework. The derivations follow the usual route of requiring the fixed-point equation to respect the inequalities at the knots and between them, and the citations to prior FIF and HVIFS papers are appropriate. The examples confirm that when the alphas satisfy the bounds the resulting curves do what is claimed. The open question is whether the intersection of those inequalities is nonempty for every choice of interpolation points and values while also keeping the overall map contractive. The zipper and hidden-variable maps can tighten the allowable range for alpha beyond the usual absolute-value-less-than-one rule, and positivity plus slope preservation can pull in different directions. The paper demonstrates feasibility only on selected data; it does not contain a general existence argument. That is a real but contained limitation rather than a flaw in the derivations themselves. Readers already working on shape-constrained fractal approximation or IFS-based fitting will find the new parameter set and the explicit conditions useful. The work is concrete enough and the claims are checkable enough that it should go to referees rather than be desk-rejected.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a Zipper Hidden Variable Iterated Function System (ZHVIFS) and constructs univariate Zipper Fractal Interpolation Functions (ZFIFs) based on it. It derives conditions on the vertical scaling factors to ensure the ZFIF preserves the boundedness, positivity, and piecewise slope of the interpolation data set, with verification through examples.

Significance. This approach offers increased flexibility in fractal interpolation by combining zipper and hidden variable features, allowing for shape preservation in a broader class of functions. The explicit conditions on scaling factors, if they admit solutions in general, represent a useful advancement for applications requiring preservation of data properties like positivity and monotonicity in fractal models.

major comments (2)

The conditions on vertical scaling factors for positivity and piecewise slope preservation are stated, but it is not shown that these inequalities always have a non-empty intersection that also satisfies the contractivity condition of the ZHVIFS (typically |α_i| < 1). The examples only cover specific cases where suitable factors exist.
The definition of the ZHVIFS and the associated maps should explicitly include the contractivity conditions to ensure the attractor exists as a continuous function; the interaction with the new parameters needs clarification to confirm the IFS is contractive under the derived bounds.

minor comments (1)

Typo in abstract: 'piecewise slop' should be 'piecewise slope'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address the major comments point by point below, indicating the revisions that will be incorporated into the next version of the manuscript.

read point-by-point responses

Referee: The conditions on vertical scaling factors for positivity and piecewise slope preservation are stated, but it is not shown that these inequalities always have a non-empty intersection that also satisfies the contractivity condition of the ZHVIFS (typically |α_i| < 1). The examples only cover specific cases where suitable factors exist.

Authors: We agree that the manuscript does not establish a general result guaranteeing that the derived inequalities for positivity and piecewise slope preservation always possess a non-empty intersection satisfying the contractivity condition |α_i| < 1. The examples demonstrate existence in specific cases, but a general existence theorem or sufficient conditions for the intersection to be non-empty are absent. In the revision we will add a remark clarifying that the choice of scaling factors must be made to satisfy all constraints simultaneously and will include a brief discussion of how to select parameters (for instance, by taking sufficiently small |α_i| when the data permit) along with an additional numerical example illustrating the selection process. We do not claim the intersection is always non-empty for arbitrary data. revision: yes
Referee: The definition of the ZHVIFS and the associated maps should explicitly include the contractivity conditions to ensure the attractor exists as a continuous function; the interaction with the new parameters needs clarification to confirm the IFS is contractive under the derived bounds.

Authors: We accept this observation. The current definition of the ZHVIFS states the maps but does not explicitly list the contractivity requirement on the vertical scaling factors in the opening definition. In the revised manuscript we will amend the definition of the ZHVIFS to include the explicit condition that the vertical scaling factors satisfy |α_i| < 1 (with respect to the metric used on the space) and will add a short paragraph explaining how the zipper and hidden-variable parameters interact with this bound to guarantee that the IFS remains contractive, thereby ensuring the existence of a unique continuous attractor. revision: yes

Circularity Check

0 steps flagged

No significant circularity in ZHVIFS-based ZFIF construction and preservation conditions

full rationale

The paper introduces a zipper hidden variable IFS and constructs the associated univariate ZFIFs as a direct extension of standard IFS fixed-point theory. It then derives explicit inequalities on the vertical scaling factors that enforce boundedness, positivity, and piecewise-slope preservation by substituting the interpolation conditions into the attractor equation. This is a conventional forward derivation from the contractivity requirement and the data constraints; no step reduces the claimed result to a fitted parameter, a self-definition, or a load-bearing self-citation. The abstract frames the work as building on general IFS machinery with new parameters whose admissible range is characterized mathematically rather than assumed or renamed. Consequently the central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The abstract provides no explicit list of axioms or free parameters; the ledger is therefore populated from the minimal claims visible in the abstract. The vertical scaling factors function as adjustable parameters whose values must be chosen to meet preservation inequalities.

free parameters (1)

vertical scaling factors
Parameters whose magnitudes must be restricted to ensure the interpolation preserves boundedness, positivity, and piecewise slope; their specific values are not given but are central to the stated conditions.

axioms (1)

domain assumption A zipper hidden variable iterated function system (ZHVIFS) can be defined and iterated to produce a continuous interpolation function.
Invoked when the paper states that ZFIFs are constructed using the ZHVIFS.

invented entities (1)

Zipper Hidden Variable Iterated Function System (ZHVIFS) no independent evidence
purpose: To generate a more general class of fractal interpolation functions with greater shape diversity than standard FIFs or HVIFS.
Newly introduced construct whose properties are used to build the ZFIFs; no independent existence proof or external reference is supplied in the abstract.

pith-pipeline@v0.9.0 · 5395 in / 1409 out tokens · 46760 ms · 2026-05-10T19:43:00.685298+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 find the conditions on vertical scaling factors for the ZFIFs to preserve boundedness, positivity and piecewise slop of the data set
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

If 1 < S, then there exists a distance θρ equivalent to the Euclidean metric such that ω_i are contractive

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

27 extracted references · 27 canonical work pages

[1]

Assev, A.V

V.V. Assev, A.V. Tetenov, A.S. Kravchenko, On self-similar Jordan curves on the plane. Sib. Math. J., 44(3) (2003) 379–386

work page 2003
[2]

Attia, M

N. Attia, M. Balegh, R. Amami, R. Amami, On the Fractal interpolation functions associated with Matkowski contractions, Electronic Research Archive, 31(8) (2023) 4652–4668

work page 2023
[3]

Balázs, K

B. Balázs, K. Gergely, K. István, Pointwise regularity of parameterized affine zipper fractal curves, Nonlinearity, 31 (2018) 1705–1733

work page 2018
[4]

Barnsley, Fractal functions and interpolation, Constr

M.F. Barnsley, Fractal functions and interpolation, Constr. Approx., 2 (1986) 303–329

work page 1986
[5]

Barnsley, J.H

M.F. Barnsley, J.H. Elton, Recurrent iterated function systems, Constr. Approx., 5 (1989) 3-31

work page 1989
[6]

Barnsley, D

M.F. Barnsley, D. Hardin, P. Massopust, Hidden variable fractal interpolation functions, SIAM J. Math. Anal., 20 (1989) 1218-1242

work page 1989
[7]

Bouboulis, L

P. Bouboulis, L. Dalla, Hidden variable vector valued fractal interpolation functions, Fractals, 13(3) (2005) 227–232

work page 2005
[8]

Bouboulis, L

P. Bouboulis, L. Dalla, V. Drakopoulos, Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension, J. Approx. Th., 141 (2006), 99-117

work page 2006
[9]

Chand, N.Vijender, P

A.K.B. Chand, N.Vijender, P. Viswanathan, A. V. Tetenov, Affine zipper fractal interpolation functions, BIT, 60(2) (2020) 319–344

work page 2020
[10]

Kapoor, S.A

G.P. Kapoor, S.A. Prasad, Smoothness of hidden variable bivariate coalescence fractal interpolation surfaces, Int. J. Bifurc, Chaos, 19(7) (2009) 2321–2333

work page 2009
[11]

Kapoor, S.A

G.P. Kapoor, S.A. Prasad, Stability of Coalescence Hidden Variable Fractal Interpolation Surfaces, Int. J. Nonlinear Sci., 9(3) (2010) 265-275

work page 2010
[12]

Katiyar, A.K.B

S.K. Katiyar, A.K.B. Chand, G.Saravana Kumar, A new class of rational cubic spline fractal interpolation function and its constrained aspects, Appl.Math.Comput., 346 (2019) 319-335

work page 2019
[13]

Kim, H.M

J.M. Kim, H.M. Mun, Nonlinear recurrent hidden variable fractal interpolation curves with function vertical scaling factors, Fractals, 28(2020) 2050096

work page 2020
[14]

Kim, H.J

J.M. Kim, H.J. Kim, H.M. Mun, Nonlinear fractal interpolation curves with function vertical scaling factors. Indian J. Pure Appl. Math., 51(2) (2020) 483-499

work page 2020
[15]

Kim, J.M

H.J. Kim, J.M. Kim, H.M. Mun, New nonlinear recurrent hidden variable fractal interpolation surfaces, Fractals, 28(2) (2020) 2050038

work page 2020
[16]

Metzler, C.H

W. Metzler, C.H. Yun, Construction of fractal interpolation surfaces on rectangular grids, Int. J. of Bifurcat. Chaos, 20(12) (2010), 4079–4086

work page 2010
[17]

M. Radu, R. Pasupathi, Contractive Multivariate Zipper Fractal Interpolation Functions, Results Math., 79(4) (2024) DOI: 10.1007/s00025-024-02177-5

work page doi:10.1007/s00025-024-02177-5 2024
[18]

Ri, New types of fractal interpolation surfaces, Chaos Solitons Fractals, 119 (2019) 291–297

S. Ri, New types of fractal interpolation surfaces, Chaos Solitons Fractals, 119 (2019) 291–297

work page 2019
[19]

M.G. Ri, C.H. Yun, Smoothness and fractional integral of hidden variable recurrent fractal interpolation function with function vertical scaling factors, Fractals, 29(6) (2021) 2150136, 1-17

work page 2021
[20]

M.G. Ri, C.H. Yun, M.H. Kim, Construction of cubic spline hidden variable recurrent fractal interpolation function and its fractional calculus, Chaos Solitons Fractals, 150 (2021) 111177, 1-11

work page 2021
[21]

M.G. Ri, C.H. Yun, Smoothness and fractional integral of hidden variable recurrent fractal interpolation function with function vertical scaling factors, Fractals, 29(6) (2021), 2150136, 1-17

work page 2021
[22]

Vijay , A.K.B

N.V. Vijay , A.K.B. Chand, C1-Positivity preserving Bi-quintic blended rational quartic zipper fractal interpolation surfaces, Chaos Solitons Fractals, 188 (2024) 115472, DOI: https://doi.org/10.1016/j.chaos.2024.115472

work page doi:10.1016/j.chaos.2024.115472 2024
[23]

Vijay, M.G.P

N.V. Vijay, M.G.P. Prasad, Gurunathan, A novel class of zipper fractal Bezier curves and its graphics applications, Chaos Solitons Fractals, 190 (2024) 115793, DOI: https://doi.org/10.1016/j.chaos.2024.115793

work page doi:10.1016/j.chaos.2024.115793 2024
[24]

Vijay, A.K.B

N.V. Vijay, A.K.B. Chand, Generalized zipper fractal approximation and parameter identification problems, Comp. App. Math., 41 (2022) 155, 1-23

work page 2022
[25]

Yun, H.C

C.H. Yun, H.C. Choi, H.C. O, Construction of recurrent fractal interpolation surfaces with function vertical scaling factors and estimation of box-counting dimension on rectangular grids, Fractals, 23(3) (2015) 1-10

work page 2015
[26]

Yun, M.G

C.H. Yun, M.G. Ri, Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors, Chaos Solitons Fractals, 134 (2020) 109700, 1-10

work page 2020
[27]

Yun, M.G

C.H. Yun, M.G. Ri, Analytic properties of hidden variable bivariable fractal interpolation functions with four function contractivity factors, Fractals, 27(2) (2019) 1950018, 1-16

work page 2019

[1] [1]

Assev, A.V

V.V. Assev, A.V. Tetenov, A.S. Kravchenko, On self-similar Jordan curves on the plane. Sib. Math. J., 44(3) (2003) 379–386

work page 2003

[2] [2]

Attia, M

N. Attia, M. Balegh, R. Amami, R. Amami, On the Fractal interpolation functions associated with Matkowski contractions, Electronic Research Archive, 31(8) (2023) 4652–4668

work page 2023

[3] [3]

Balázs, K

B. Balázs, K. Gergely, K. István, Pointwise regularity of parameterized affine zipper fractal curves, Nonlinearity, 31 (2018) 1705–1733

work page 2018

[4] [4]

Barnsley, Fractal functions and interpolation, Constr

M.F. Barnsley, Fractal functions and interpolation, Constr. Approx., 2 (1986) 303–329

work page 1986

[5] [5]

Barnsley, J.H

M.F. Barnsley, J.H. Elton, Recurrent iterated function systems, Constr. Approx., 5 (1989) 3-31

work page 1989

[6] [6]

Barnsley, D

M.F. Barnsley, D. Hardin, P. Massopust, Hidden variable fractal interpolation functions, SIAM J. Math. Anal., 20 (1989) 1218-1242

work page 1989

[7] [7]

Bouboulis, L

P. Bouboulis, L. Dalla, Hidden variable vector valued fractal interpolation functions, Fractals, 13(3) (2005) 227–232

work page 2005

[8] [8]

Bouboulis, L

P. Bouboulis, L. Dalla, V. Drakopoulos, Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension, J. Approx. Th., 141 (2006), 99-117

work page 2006

[9] [9]

Chand, N.Vijender, P

A.K.B. Chand, N.Vijender, P. Viswanathan, A. V. Tetenov, Affine zipper fractal interpolation functions, BIT, 60(2) (2020) 319–344

work page 2020

[10] [10]

Kapoor, S.A

G.P. Kapoor, S.A. Prasad, Smoothness of hidden variable bivariate coalescence fractal interpolation surfaces, Int. J. Bifurc, Chaos, 19(7) (2009) 2321–2333

work page 2009

[11] [11]

Kapoor, S.A

G.P. Kapoor, S.A. Prasad, Stability of Coalescence Hidden Variable Fractal Interpolation Surfaces, Int. J. Nonlinear Sci., 9(3) (2010) 265-275

work page 2010

[12] [12]

Katiyar, A.K.B

S.K. Katiyar, A.K.B. Chand, G.Saravana Kumar, A new class of rational cubic spline fractal interpolation function and its constrained aspects, Appl.Math.Comput., 346 (2019) 319-335

work page 2019

[13] [13]

Kim, H.M

J.M. Kim, H.M. Mun, Nonlinear recurrent hidden variable fractal interpolation curves with function vertical scaling factors, Fractals, 28(2020) 2050096

work page 2020

[14] [14]

Kim, H.J

J.M. Kim, H.J. Kim, H.M. Mun, Nonlinear fractal interpolation curves with function vertical scaling factors. Indian J. Pure Appl. Math., 51(2) (2020) 483-499

work page 2020

[15] [15]

Kim, J.M

H.J. Kim, J.M. Kim, H.M. Mun, New nonlinear recurrent hidden variable fractal interpolation surfaces, Fractals, 28(2) (2020) 2050038

work page 2020

[16] [16]

Metzler, C.H

W. Metzler, C.H. Yun, Construction of fractal interpolation surfaces on rectangular grids, Int. J. of Bifurcat. Chaos, 20(12) (2010), 4079–4086

work page 2010

[17] [17]

M. Radu, R. Pasupathi, Contractive Multivariate Zipper Fractal Interpolation Functions, Results Math., 79(4) (2024) DOI: 10.1007/s00025-024-02177-5

work page doi:10.1007/s00025-024-02177-5 2024

[18] [18]

Ri, New types of fractal interpolation surfaces, Chaos Solitons Fractals, 119 (2019) 291–297

S. Ri, New types of fractal interpolation surfaces, Chaos Solitons Fractals, 119 (2019) 291–297

work page 2019

[19] [19]

M.G. Ri, C.H. Yun, Smoothness and fractional integral of hidden variable recurrent fractal interpolation function with function vertical scaling factors, Fractals, 29(6) (2021) 2150136, 1-17

work page 2021

[20] [20]

M.G. Ri, C.H. Yun, M.H. Kim, Construction of cubic spline hidden variable recurrent fractal interpolation function and its fractional calculus, Chaos Solitons Fractals, 150 (2021) 111177, 1-11

work page 2021

[21] [21]

M.G. Ri, C.H. Yun, Smoothness and fractional integral of hidden variable recurrent fractal interpolation function with function vertical scaling factors, Fractals, 29(6) (2021), 2150136, 1-17

work page 2021

[22] [22]

Vijay , A.K.B

N.V. Vijay , A.K.B. Chand, C1-Positivity preserving Bi-quintic blended rational quartic zipper fractal interpolation surfaces, Chaos Solitons Fractals, 188 (2024) 115472, DOI: https://doi.org/10.1016/j.chaos.2024.115472

work page doi:10.1016/j.chaos.2024.115472 2024

[23] [23]

Vijay, M.G.P

N.V. Vijay, M.G.P. Prasad, Gurunathan, A novel class of zipper fractal Bezier curves and its graphics applications, Chaos Solitons Fractals, 190 (2024) 115793, DOI: https://doi.org/10.1016/j.chaos.2024.115793

work page doi:10.1016/j.chaos.2024.115793 2024

[24] [24]

Vijay, A.K.B

N.V. Vijay, A.K.B. Chand, Generalized zipper fractal approximation and parameter identification problems, Comp. App. Math., 41 (2022) 155, 1-23

work page 2022

[25] [25]

Yun, H.C

C.H. Yun, H.C. Choi, H.C. O, Construction of recurrent fractal interpolation surfaces with function vertical scaling factors and estimation of box-counting dimension on rectangular grids, Fractals, 23(3) (2015) 1-10

work page 2015

[26] [26]

Yun, M.G

C.H. Yun, M.G. Ri, Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors, Chaos Solitons Fractals, 134 (2020) 109700, 1-10

work page 2020

[27] [27]

Yun, M.G

C.H. Yun, M.G. Ri, Analytic properties of hidden variable bivariable fractal interpolation functions with four function contractivity factors, Fractals, 27(2) (2019) 1950018, 1-16

work page 2019