# Scale-Free Fractal Interpolation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (i)
- φ-contraction if$$d\left(f\right(x),f(y\left)\right)\le \phi \left(d\right(x,y\left)\right)$$
- (ii)
- Rakotch contraction if it is a φ-contraction where the function φ satisfies: $t\to \frac{\phi \left(t\right)}{t}$ is non-increasing and $\frac{\phi \left(t\right)}{t}<1$, for every $t>0$.
- (iii)
- Matkowski contraction if it is a φ-contraction where the function φ is non-decreasing and the following limit holds $\underset{n\to \infty}{lim}{\phi}^{n}\left(t\right)=0$, for every $t>0$.

**Remark**

**1.**

**Remark**

**2.**

**Theorem**

**1.**

**Definition**

**2.**

- (i)
- J is finite;
- (ii)
- ${f}_{n}:X\to X$ are continuous functions;
- (iii)
- $(X,d)$ is a complete metric space.

**Remark**

**3.**

**Definition**

**3.**

## 3. Fractal Interpolation Associated with Matkowski Contractions

#### 3.1. Finite Number of Data

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Definition**

**4.**

**Corollary**

**1.**

**Proof.**

**Remark**

**4.**

**Example**

**1.**

#### 3.2. Infinite Number of Data

**Theorem**

**3.**

**Proof.**

**Definition**

**5.**

**Corollary**

**2.**

**Proof.**

**Remark**

**5.**

**Remark**

**6.**

**Remark**

**7.**

## 4. R-Fractal Interpolation Functions

**Definition**

**6.**

**Remark**

**8.**

**Remark**

**9.**

**Remark**

**10.**

**Theorem**

**4.**

**Proof.**

**Remark**

**11.**

#### 4.1. Properties of R-Fractal Interpolation Functions

**Proposition**

**1.**

**Proof.**

#### 4.2. Case ${S}_{n}={R}_{n}\circ B$

#### 4.3. Linear Case

## 5. Smooth R-Fractal Interpolation Functions

**Theorem**

**5.**

**Proof.**

**Remark**

**12.**

**Theorem**

**6.**

**Proof.**

**Remark**

**13.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Navascués, M.A.; Pacurar, C.; Drakopoulos, V.
Scale-Free Fractal Interpolation. *Fractal Fract.* **2022**, *6*, 602.
https://doi.org/10.3390/fractalfract6100602

**AMA Style**

Navascués MA, Pacurar C, Drakopoulos V.
Scale-Free Fractal Interpolation. *Fractal and Fractional*. 2022; 6(10):602.
https://doi.org/10.3390/fractalfract6100602

**Chicago/Turabian Style**

Navascués, María A., Cristina Pacurar, and Vasileios Drakopoulos.
2022. "Scale-Free Fractal Interpolation" *Fractal and Fractional* 6, no. 10: 602.
https://doi.org/10.3390/fractalfract6100602