Zipper Rational Quadratic Fractal Interpolation Functions
01 Jan 2021-pp 229-241
TL;DR: In this article, the authors proposed an interpolation method using a binary parameter called signature such that the graph of the interpolant is an attractor of a suitable zipper rational iterated function system.
Abstract: Inthis article, we propose an interpolation method using a binary parameter called signature such that the graph of the interpolant is an attractor of a suitable zipper rational iterated function system. The presence of scaling factors and signature in the proposed zipper rational quadratic fractal interpolation functions (ZRQFIFs) gives the flexibility to produce a wide variety of interpolants. Using suitable conditions on the scale factor and shape parameter, we construct a \(\mathcal {C}^1\)-continuous ZRQFIF from a \(\mathcal {C}^0\)-continuous ZRQFIF. We also establish the uniform convergence of ZRQFIF to an original data-generating function. Further, we deduce suitable conditions on the IFS parameters and shape parameters to retain the positivity feature associated with a prescribed data by the proposed interpolant.
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TL;DR: In this article, the authors introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x��i)=y fixmei fori e {0,1,⋯,N}.
Abstract: Let a data set {(x
i,y
i) ∈I×R;i=0,1,⋯,N} be given, whereI=[x
0,x
N]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x
i)=y
i fori e {0,1,⋯,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.
736 citations
Book•
01 Jan 1994
TL;DR: Fractal Function Wavelet theory as mentioned in this paper is a well-known extension of the basic wavelet theory and has been applied to the construction of Fractal Sets as Fractal Functions and Fractal Surfaces.
Abstract: (Subchapter Titles): I. Foundations. Mathematical Preliminaries: Analysis and Topology. Probability Theory. Algebra. Construction of Fractal Sets: Classical Fractal Sets. Iterated Function Systems. Recurrent Sets. Graph Directed Fractal Constructions. Dimension Theory: Topological Dimensions. Metric Dimensions. Probabilistic Dimensions. Dimension Results for Self-Affine Fractals. The Box Dimension of Projections. Dynamical Systems and Dimension. II. Fractal Functions and Fractal Surfaces: Fractal Function Construction: The Read-BajraktarevicOperator. Recurrent Sets as Fractal Functions. Iterative Interpolation Functions. Recurrent Fractal Functions. Hidden Variable Fractal Functions. Properties of Fractal Functions. Peano Curves. Fractal Functions of Class C gt gt . Dimension of Fractal Functions: gt Dimension Calculations. Function Spaces and Dimension. Fractal Functions and Wavelets: gt Basic Wavelet Theory. Fractal Function Wavelets. Fractal Surfaces: gt Tensor Product Fractal Surfaces. Affine Fractal Surfaces in R gt n+M gt . Properties of Fractal Surfaces. Fractal Surfaces of Class Ck gt . Fractal Surfaces and Wavelets in R gt n gt : gt Brief Review of Coxeter Groups. Fractal Functions on Foldable Figures. Interpolation on Foldable Figures. Dilation and W gt Invariant Spaces. Multiresolution Analyses. List of Symbols. Bibliography. Author Index. Subject Index.
239 citations
TL;DR: The calculus of deterministic fractal functions is introduced in this article, which can be explicitly indefinitely integrated any number of times, yielding a hierarchy of successively smoother interpolation functions which generalize splines and which are attractors for iterated function systems.
237 citations
TL;DR: In this article, the authors constructed interpolation functions of the form f[0, 1] \to \mathbb{R}$ of the following nature: given data, f obeys the following properties: f(t_n ) = x_n,\qquad n = 0,1,2, \cdots,N.
Abstract: Interpolation functions $f:[0,1] \to \mathbb{R}$ of the following nature are constructed. Given data \[ \left\{ {\left( {t_n ,x_n } \right) \in [0,1] \times \mathbb{R}:n = 0,1,2, \cdots ,N} \right\}\] with $0 = t_0 < t_1 < \cdots <' t_N = 1$, f obeys \[ f(t_n ) = x_n ,\qquad n = 0,1,2, \cdots ,N.\] Furthermore, the graph of f is the projection of a set G in $\mathbb{R}^M $ (M an integer greater than or equal to 2) that is homeomorphic to $[0,1]$ and is the attractor for an iterated function system consisting of afflne maps in $\mathbb{R}^M $. The latter characterization ensures that f can be computed rapidly while possessing many ”hidden” variables, on which its values continuously depend, which allow great flexibility and diversity in the interpolant, making it potentially useful in approximation theory. Estimtes and exact values for the fractal dimensions of G and the graph of f are obtained.
188 citations