Generalised Vector-Valued Fractal Interpolation

• The paper introduces a method for constructing continuous multi-dimensional fractal interpolants via iterated function systems using variable scaling and hidden variables.
• The methodology employs affine partition maps and nonlinear vertical maps under contractive Edelstein conditions to ensure existence, uniqueness, and smoothness of the solution.
• The framework bridges self-affine and non-self-affine interpolation, offering tunable fractal dimension bounds and applications in image compression, forecasting, and geometric design.

A generalised vector-valued fractal interpolation function (FIF) is a continuous function $F: I \to \mathbb{R}^m$ whose graph is constructed as the attractor of an iterated function system (@@@@1@@@@) equipped with variable scaling factors and hidden variables, generalizing both the classical scalar FIF and extending flexibility in interpolation and shape control. Such constructions incorporate advanced contractivity conditions—Edelstein contractions and graph-directed coalescence hidden-variable schemes—enabling interpolation of multi-dimensional or coupled datasets, smoothness characterization, and nontrivial fractal dimension bounds. The existence, uniqueness, and continuity of these interpolants follow from fixed-point arguments in appropriate functional spaces, encompassing both self-affine and non-self-affine behaviors according to parameter regimes (T et al., 22 Jan 2026, Akhtar et al., 2015).

1. Data Structures and Hidden Variable Lifting

Construction begins with a finite collection of data points:

• For scalar and vector settings, one considers $x_0 < x_1 < \cdots < x_n$ in %%%%2%%%% and interpolation values $y_0,\ldots,y_n \in \mathbb{R}^m$.
• To enable greater flexibility, each data point can be lifted to $\mathbb{R}^{m+1}$ by adjoining a hidden variable: e.g., $(x_i, y_i, z_i)$ for $m=2$; $z_i$ is freely chosen subject to mild join-up conditions.
• In the graph-directed setting, multiple such data sets can be simultaneously interpolated, with each set associated to a vertex in a directed graph (Akhtar et al., 2015).

This lifting introduces "shape parameters" and coupling possibilities that ultimately govern self-affinity, non-self-affinity, and the geometric complexity of the resulting interpolant.

2. Iterated Function Systems and Edelstein/Graph-Directed Contractions

The general IFS framework encompasses both classical affine contractions and generalised, component-wise nonlinear contractions:

• Affine Partition Maps: For each interval $I=[x_0,x_n]$, subintervals $I_i=[x_{i-1},x_i]$ are associated to affine maps $h_i(x)=a_i x + b_i$ with $0<a_i<1$ such that $h_i(x_0)=x_{i-1}$ and $h_i(x_n)=x_i$.
• Hidden-Variable/Nonlinear Vertical Maps: For vector-valued scenarios ($m>1$), vertical scaling is effected through functions $S_i: \mathbb{R}^m \to \mathbb{R}^m$, defined as $S_i(u) = (s_{i,1}(u_1), \ldots, s_{i,m}(u_m))^T$, with each $s_{i,j}$ an Edelstein contraction: $|s_{i,j}(u') - s_{i,j}(u'')| < |u' - u''|$ for $u'\neq u''$.
• Matrix Coupling and Join-Up Data: Nonconstant matrix-valued $D_i(x)\in C(I;M_{m\times m})$ and vector-valued $Q_i(x)\in C(I; \mathbb{R}^m)$ provide additional adjustment. The maps are defined as $F_i(x,u) = D_i(x) S_i(u) + Q_i(x)$.

In the graph-directed setting, the IFS consists of a family of maps parameterized by directed edges and their multiplicities, $w_n^{rs}: X^s \rightarrow X^r$, with composition defined for graphs $G=(V,E)$ (Akhtar et al., 2015). The contractivity conditions are critical: Edelstein contractivity in the nonlinear case and norm bounds for matrix components (e.g., $\|D_i(x)\|_\text{op} \leq \theta_i < 1$).

3. Functional Equation and Fixed Point Existence

The interpolant function $f$ is defined as the unique fixed point of a Read–Bajraktarević (R–B) operator acting on the space of continuous functions $C_d(I) = \{f\in C(I;\mathbb{R}^m): f(x_i) = y_i\}$:

$$(\mathcal{R}f)(x) = F_i(h_i^{-1}(x), f(h_i^{-1}(x))) \quad \text{for } x \in I_i.$$

This yields the functional recursion:

$$(3.1)\quad f(x) = D_i(h_i^{-1}(x)) S_i(f(h_i^{-1}(x))) + Q_i(h_i^{-1}(x)).$$

The contractivity of $w_i$ on $(x,u)$ and the Edelstein contraction of $\mathcal{R}$ on $C_d(I)$ ensures—via the Edelstein fixed-point theorem—that a unique continuous interpolant exists (T et al., 22 Jan 2026). For the graph-directed scenario, similar Banach–Hutchinson arguments apply, leading to unique attractors $G^r$ for each dataset, with the projected graphs delivering the CHFIF structure (Akhtar et al., 2015).

4. Self-Affinity, Non-Self-Affinity, and Coupling Structure

A key feature is the ability to transition between self-affine and non-self-affine interpolation:

• Setting coupling/mixing parameters (e.g., $\beta_n^{rs}$ in coalescence CHFIF constructions) to zero yields a classical self-affine structure where each component evolves independently.
• Nonzero parameters admit non-self-affine phenomena, with cross-coupling among hidden variables and interpolant components. This models more complex datasets and allows tailored fractal characteristics, such as roughness and multi-scale behavior (Akhtar et al., 2015, T et al., 22 Jan 2026).
• The matrix coupling $D_i(x)$ and nonlinear $S_i$ generalize the vertical scaling beyond constant coefficients, and support arbitrarily nonlinear interpolation—subject to contractivity constraints.

5. Smoothness and Hölder Continuity

Iterating the functional equation allows explicit estimates for Hölder continuity:

• Let $\lambda = \max_i a_i < 1$, $\theta = \max_i \sup_{x\in I} \|D_i(x)\| < 1$, and suppose each $Q_i$ is $L_q$-Lipschitz.
• Then for any $\alpha$ with $\theta \lambda^{-\alpha}<1$, the interpolant satisfies

$\|f(x) - f(x')\| \leq K |x - x'|^\alpha,$

for all $x,x' \in I$ and some $K>0$ (T et al., 22 Jan 2026). The exponent $\alpha$ embodies the interplay between horizontal and vertical contraction, with large vertical oscillations ($\theta\rightarrow 1$) driving $\alpha\rightarrow 0$ and maximizing graph roughness.

6. Fractal Dimension and Box-Counting Estimates

Once $\alpha$-Hölder continuity is established, box-counting dimension bounds can be obtained:

• The basic estimate is $\dim_B\{(x, f(x)): x \in I\} \leq 2 - \alpha$.
• Tighter bounds follow from pressure-type equations. For linear components $\{a_i, \theta_i\}$, one computes $D^* \leq D$ where $D^*$ solves

$\sum_{i=1}^n a_i^{D^* - 1} \theta_i^{D^*} = 1.$

Equal scaling $a_i=a$, $\theta_i=\theta$ yields $D^* = 1 - \frac{\ln \theta}{\ln a}$ (T et al., 22 Jan 2026). In the graph-directed CHFIF setting with self-affinity, the box or Hausdorff dimension $D$ solves

$\sum_{i=1}^N |a_i|^D |\alpha_i| = 1,$

while for non-self-affine parameter regimes, singular-value function techniques are required (Akhtar et al., 2015).

7. Generalizations, Applications, and Framework Unification

The generalised vector-valued FIF construction encapsulates and broadens earlier scalar and affine methodologies:

• Classical FIFs [Barnsley]: recovered when $D_i(x)\equiv d_i$, $S_i(u)=u$, $Q_i$ affine (T et al., 22 Jan 2026).
• Added "hidden-variable" vertical scaling and position-dependent matrix couplings enhance flexibility, modeling a wide class of self-similar and nonlinear phenomena: fractal-based image compression, time-series forecasting, computer-aided geometric design, and PDE solution representation via fractal-Galerkin techniques.
• The condition $\theta < \lambda^{\alpha}$ is both necessary for $\alpha$-Hölder continuity and governs fractal dimension—establishing a tradeoff between interpolant smoothness and graph complexity (T et al., 22 Jan 2026, Akhtar et al., 2015).

The unified framework thus delivers rich families of vector-valued fractal interpolants with explicit control of continuity, differentiability, and dimension, founded upon rigorous IFS theory and generalised contractivity conditions.