# Smooth fractal interpolation

@article{Navascus2006SmoothFI, title={Smooth fractal interpolation}, author={Mar{\'i}a Antonia Navascu{\'e}s and Mar{\'i}a Victoria Sebasti{\'a}n}, journal={Journal of Inequalities and Applications}, year={2006}, volume={2006}, pages={1-20} }

Fractal methodology provides a general frame for the understanding of real-world phenomena. In particular, the classical methods of real-data interpolation can be generalized by means of fractal techniques. In this paper, we describe a procedure for the construction of smooth fractal functions, with the help of Hermite osculatory polynomials. As a consequence of the process, we generalize any smooth interpolant by means of a family of fractal functions. In particular, the elements of the class…

## 78 Citations

### MULTIVARIATE AFFINE FRACTAL INTERPOLATION

- Mathematics
- 2020

Fractal interpolation functions capture the irregularity of some data very effectively in comparison with the classical interpolants. They yield a new technique for fitting experimental data sampled…

### Fractal interpolation: a sequential approach

- MathematicsApplied Mathematics-A Journal of Chinese Universities
- 2021

Fractal interpolation is a modern technique to fit and analyze scientific data. We develop a new class of fractal interpolation functions which converge to a data generating (original) function for…

### Fractal-classic interpolants

- Mathematics
- 2009

The methodology of fractal interpolation is very useful for processing experimental signals in order to extract their characteristics of complexity. We go further and prove that the Iterated Function…

### Fractal Approximants on the Circle

- Computer Science, Mathematics
- 2018

In this article, a generalization of some approximation formulae proposed by Dunham Jackson both in classical and fractal cases are studied.

### A NEW CLASS OF FRACTAL INTERPOLATION SURFACES BASED ON FUNCTIONAL VALUES

- Mathematics
- 2016

Fractal interpolation is a modern technique for fitting of smooth/non-smooth data. Based on only functional values, we develop two types of 𝒞1-rational fractal interpolation surfaces (FISs) on a…

### FRACTAL APPROXIMATION OF JACKSON TYPE FOR PERIODIC PHENOMENA

- MathematicsFractals
- 2018

The reconstruction of an unknown function providing a set of Lagrange data can be approached by means of fractal interpolation. The power of that methodology allows us to generalize any other…

### A fractal procedure for the computation of mixed interpolants

- Mathematics
- 2008

SUMMARY We develop a procedure from the fractal methodology for the computation of an interpolant born from the cooperation of two functions of dierent nature. In particular, we define an Iterated…

### Fundamental Sets of Fractal Functions

- Mathematics
- 2008

Abstract
Fractal interpolants constructed through iterated function systems prove more general than classical interpolants. In this paper, we assign a family of fractal functions to several classes…

### Toward a Unified Methodology for Fractal Extension of Various Shape Preserving Spline Interpolants

- Mathematics
- 2015

Fractal interpolation, one in the long tradition of those involving the interpolatary theory of functions, is concerned with interpolation of a data set with a function whose graph is a fractal or a…

### Visualization of constrained data by smooth rational fractal interpolation

- MathematicsInt. J. Comput. Math.
- 2016

A family of smooth rational spline fractal interpolation function (SRFIF) is presented with the help of classical rational cubic spline and convergence analysis shows that SRFIF gives a good approximation to the original function.

## References

SHOWING 1-10 OF 16 REFERENCES

### Some results of convergence of cubic spline fractal interpolation functions.

- Mathematics
- 2002

Fractal interpolation functions (FIFs) provide new methods of approximation of experimental data. In the present paper, a fractal technique generalizing cubic spline functions is proposed. A FIF f is…

### A Fractal Version of the Schultz's Theorem

- Mathematics
- 2005

The approximation of experimental data can be envisaged in the light of fractal
interpolation functions defined by iterated function systems. In the particular
case of polynomial fractal…

### Generalized Cubic Spline Fractal Interpolation Functions

- MathematicsSIAM J. Numer. Anal.
- 2006

In view of wide ranging applications of the classical cubic splines in several mathematical and engineering problems, the explicit construction of cubic spline FIF $f_{\Delta}(x)$ through moments is developed and it is shown that the sequence f_{Delta_k} (x) converges to the defining data function on two classes of sequences of meshes at least as rapidly as the square of the mesh norm approaches to zero.

### Fractal functions and interpolation

- Mathematics
- 1986

Let a data set {(xi,yi) ∈I×R;i=0,1,⋯,N} be given, whereI=[x0,xN]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data…

### Fractals everywhere

- Physics
- 1988

Focusing on how fractal geometry can be used to model real objects in the physical world, this up-to-date edition featurestwo 16-page full-color inserts, problems and tools emphasizing fractal…

### Numerical methods of high-order accuracy for nonlinear boundary value Problems

- Mathematics
- 1967

with Dirichlet boundary conditions d D~u(O) = D k u ( l ) -0, D ........ dx' O ~ k < ~ n t , (/.2) where (1.3) ~Eu (x)~ = Z ( l)J+lDJEpj(x) DJu(x)l, n==_t. i=0 Basically, the Rayleigh-Ritz-Galerkin…

### Introduction to Numerical Analysis

- Mathematics
- 2001

1. The numerical evaluation of expressions 2. Linear systems of equations 3. Interpolation and numerical differentiation 4. Numerical integration 5. Univariate nonlinear equations 6. Systems of…