Nonlinear Fractal Histopolation Function

• Nonlinear fractal histopolation function is a self-referential fractal model that exactly preserves histogram areas over prescribed subintervals.
• It is constructed as the fixed point of a nonlinear Read–Bajraktarević operator using affine domain maps and nonlinear Rakotch contractions.
• The function remains bounded and Riemann integrable despite possible discontinuities, offering a robust alternative to traditional interpolation techniques.

A nonlinear fractal histopolation function is a self-referential bounded function associated with a prescribed histogram such that its integrals over a partition match the histogram areas, while continuity is not required. In the formulation introduced in "Nonlinear Fractal Histopolation Function" (T et al., 23 Sep 2025), the function is constructed as the fixed point of a nonlinear Read–Bajraktarević operator induced by an iterated function system (IFS) built from affine domain maps and nonlinear vertical maps governed by Rakotch contractions. The resulting object is a bounded, Riemann integrable, generally noncontinuous fractal function whose closed graph is the attractor of the underlying IFS and whose area-matching property is enforced by explicit conditions on the coefficients.

1. Histopolation and the underlying data model

The basic distinction is between interpolation and histopolation. Interpolation requires pointwise fitting,

$f(t_j)=y_j,\qquad j=0,1,\dots,N,$

whereas histopolation requires area matching over prescribed subintervals. For

$I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$

with

$$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$

and histogram data

$F=\{y_1,y_2,\dots,y_N\},$

a histopolating function satisfies

$$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$

where the normalized lengths are defined by

$|I_j|=a_j(t_N-t_0).$

Accordingly, histopolation is an area-preserving analogue of interpolation rather than a point-matching one (T et al., 23 Sep 2025).

This formulation is technically significant because it removes the requirement that the approximant be continuous or interpolatory at knot points. The 2025 framework treats histogram data directly, rather than recasting the problem as interpolation of cumulative quantities. That distinction is central in the fractal setting, where the relevant fixed points may be noncontinuous and where join-up conditions are not part of the definition of the histopolation problem.

2. Self-referential construction

The construction begins with affine domain contractions

$l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$

chosen so that

$l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$

Each $l_j$ therefore maps the full interval $I$ onto the subinterval $I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$0, and the family $I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$1 forms a disjoint partition of $I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$2 (T et al., 23 Sep 2025).

The vertical maps are defined on

$I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$3

by

$I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$4

where $I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$5 are constants and $I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$6 are nonlinear Rakotch contractions. A remark in the same work also allows variable vertical scaling functions $I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$7 under corresponding bounds. The associated IFS maps are

$I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$8

and the IFS is

$I=[t_0,t_N],\qquad t_0<t_1<\cdots<t_N,$9

The function itself is obtained through the Read–Bajraktarević operator on the Banach space $$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$0 of bounded real-valued functions on $$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$1, equipped with

$$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$2

For $$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$3,

$$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$4

Its unique fixed point $$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$5 satisfies the self-referential equation

$$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$6

This fixed point is the nonlinear fractal function (T et al., 23 Sep 2025).

3. Rakotch contractions and fixed-point theory

A central structural feature of the theory is the replacement of Banach contractions by Rakotch contractions. A map $$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$7 is a $$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$8-contraction if

$$I_j=[t_{j-1},t_j)\quad (j=1,\dots,N-1),\qquad I_N=[t_{N-1},t_N],$$9

It is a Rakotch contraction if, in addition,

$F=\{y_1,y_2,\dots,y_N\},$0

Every Banach contraction is Rakotch, but not conversely. The fixed-point theorem used here states that a Rakotch contraction on a complete metric space has a unique fixed point, and the iterates converge to it (T et al., 23 Sep 2025).

For the IFS on $F=\{y_1,y_2,\dots,y_N\},$1, the metric is

$F=\{y_1,y_2,\dots,y_N\},$2

with

$F=\{y_1,y_2,\dots,y_N\},$3

Under the stated assumptions, each $F=\{y_1,y_2,\dots,y_N\},$4 is a Rakotch contraction, and the IFS therefore has a unique attractor (T et al., 23 Sep 2025).

The same contractive mechanism governs the Read–Bajraktarević operator. The key estimate is

$F=\{y_1,y_2,\dots,y_N\},$5

Hence $F=\{y_1,y_2,\dots,y_N\},$6 is a Rakotch contraction on $F=\{y_1,y_2,\dots,y_N\},$7, so it has a unique fixed point. The shift from Banach to Rakotch contractions enlarges the admissible class of nonlinearities and is one of the principal theoretical extensions of the construction.

4. Boundedness, graph attractor, and integrability

The fixed point $F=\{y_1,y_2,\dots,y_N\},$8 is bounded because $F=\{y_1,y_2,\dots,y_N\},$9 maps bounded functions to bounded functions. Its graph,

$$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$0

need not be closed, because continuity is not assumed. The closure of the graph is nevertheless characterized exactly by the IFS attractor: $$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$1 The proof proceeds by establishing both

$$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$2

where

$$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$3

Thus the closed graph attractor is the geometric realization of the nonlinear fractal histopolation function (T et al., 23 Sep 2025).

Riemann integrability is obtained under the conditions

$$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$4

The proof uses oscillation estimates and Lebesgue’s criterion. The key estimate is

$$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$5

where $$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$6. Iteration along cylinder sets yields bounds tending to zero as the depth increases, so the oscillation vanishes at almost every point. Equivalently, the discontinuities lie in a countable set $$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$7, hence in a measure-zero set. The function is therefore Riemann integrable (T et al., 23 Sep 2025).

This part of the theory is important because histopolation imposes integral constraints rather than point constraints. The construction does not merely produce a bounded self-referential object; it produces one for which the relevant subinterval integrals are well defined in the Riemann sense.

5. Exact histopolation criterion and admissible vertical scaling

The decisive characterization of the area-matching property is the histopolation criterion. The fixed point $$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$8 solves the histopolation problem if and only if

$$\int_{t_{j-1}}^{t_j} f(t)\,dt = y_j\,a_j\,(t_N-t_0),$$9

This formula identifies the constants $|I_j|=a_j(t_N-t_0).$0 as the parameters that enforce the histogram-area constraints. The self-referential structure is therefore not independent of the area data: the function is histopolating precisely when the vertical offsets satisfy this integral relation (T et al., 23 Sep 2025).

A notable novelty is the controlled allowance of vertical scaling factors greater than one. Two cases are distinguished. If

$|I_j|=a_j(t_N-t_0).$1

one chooses $|I_j|=a_j(t_N-t_0).$2 such that

$|I_j|=a_j(t_N-t_0).$3

and then requires

$|I_j|=a_j(t_N-t_0).$4

If instead

$|I_j|=a_j(t_N-t_0).$5

one requires

$|I_j|=a_j(t_N-t_0).$6

In the first case, $|I_j|=a_j(t_N-t_0).$7 is permitted while still ensuring that the composite contraction factor

$|I_j|=a_j(t_N-t_0).$8

satisfies

$|I_j|=a_j(t_N-t_0).$9

A corresponding remark allows variable scaling functions $l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$0, provided

$l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$1

satisfies the same constraints (T et al., 23 Sep 2025).

These admissibility conditions distinguish the nonlinear Rakotch-based formulation from many classical fractal interpolation and histopolation schemes, where vertical scales are typically restricted by Banach-type bounds.

6. Relation to earlier fractal histopolation and representative examples

Classical fractal interpolation functions are continuous, interpolate data points, typically require join-up conditions, and often use Banach contractions. Earlier fractal histopolation, including the Barnsley–Viswanathan line of work, solves area constraints, is not necessarily continuous, and was developed with Banach contractions. The nonlinear formulation extends that setting by constructing a nonlinear fractal histopolation function, using Rakotch contractions, allowing vertical scaling factors greater than one under precise conditions, and providing a fixed-point characterization together with an explicit integral criterion for histopolation (T et al., 23 Sep 2025).

The 2015 work "Discontinuous Fractal Functions and Fractal Histopolation" formalized a fractal histopolant as an integrable fractal function satisfying

$l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$2

for a partition $l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$3, with affine maps

$l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$4

and $l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$5. It also derived the exact area constraint

$l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$6

and established that the discontinuities of the bounded fractal function form a Lebesgue null set, so the function is Riemann integrable (Barnsley et al., 2015). Relative to that framework, the nonlinear Rakotch-based construction retains the area-preserving objective while enlarging the allowable self-referential dynamics.

The 2022 work "Fractal Interpolation over Nonlinear Partitions" introduced a nonlinear partition setting

$l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$7

with nonlinear diffeomorphisms $l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$8, and the self-referential equation

$l_j:I\to I,\qquad l_j(t)=a_j t+b_j,$9

It proved existence and uniqueness in $l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$0, and also developed $l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$1 and $l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$2 theories under explicit contractivity conditions, but it did not formulate histopolation constraints (Massopust, 2022). This suggests a broader route toward histopolation over nonlinear partitions, although the 2025 construction itself is formulated on affine interval partitions.

Two examples illustrate the scope of the nonlinear theory. For $l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$3 with partition $l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$4, one example takes

$l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$5

with

$l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$6

and variable scaling factors

$l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$7

The resulting fixed point satisfies a piecewise self-referential equation and illustrates that the framework accommodates variable vertical scaling exceeding $l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$8 (T et al., 23 Sep 2025). A second example, again on the partition $l_j(t_0)=t_{j-1},\qquad l_j(t_N)=t_j.$9 but with histogram $l_j$0, uses

$l_j$1

and the fact that $l_j$2 is a Rakotch contraction. When $l_j$3 satisfy the histopolation condition above, the fixed point is a solution to the histopolation problem (T et al., 23 Sep 2025).

Taken together, these developments place the nonlinear fractal histopolation function at the intersection of fractal interpolation, discontinuous fractal approximation, and area-preserving approximation. Its defining feature is not interpolation of nodal values but exact preservation of prescribed histogram areas through a nonlinear self-referential fixed-point mechanism.