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… 
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78 Citations
Background Citations
14
Methods Citations
7

78 Citations

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References

SHOWING 1-10 OF 16 REFERENCES

Generalization of Hermite functions by fractal interpolation

Some results of convergence of cubic spline fractal interpolation functions.

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

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…

The calculus of fractal interpolation functions

Generalized Cubic Spline Fractal Interpolation Functions

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

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

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…

Optimal error bounds for cubic spline interpolation

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

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

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…