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AUTOCOVARIANCE AND INCREMENTS OF DEVIATION OF fractal interpolation functions FOR RANDOM DATASETS

fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2018年第5期26卷

作者： Luor, Dah-Chin I Shou Univ Dept Financial & Computat Math 1Sec 1Syuecheng Rd Kaohsiung 84001 Taiwan

In this paper we consider the expectation, the autocovariance, and increments of the deviation of a fractal interpolation function f(Delta Y) corresponding to a random dataset Delta(Y) = {(x(k), Y-k) : k = 0, 1, . . ., N}. We show that the covariance of f(Delta Y)(x) and Y-i is a fractal interpolation function on I for each fixed Y-i, where I = [x(0), x(N)]. We also prove that, for a fixed x is an element of I, the covariance of f Delta(Y) (x) and f(Delta Y) (t) is a fractal interpolation function on I. A special type of increments of the deviation of f(Delta Y) is also investigated.

关键词： fractals interpolation fractal interpolation functions

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STATISTICAL PROPERTIES OF LINEAR fractal interpolation functions FOR RANDOM DATA SETS

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2018年第1期26卷 1850009-1850009页

作者： Luor, Dah-Chin I Shou Univ Dept Financial & Computat Math 1Sec 1Syuecheng Rd Kaohsiung 84001 Taiwan

Let N be an integer greater than or equal to 2 and let x'(i)s be numbers with x(0) < x(1) < x(2) < ...< x(N). Denote that I is the interval left perpendicularx(0), x(N)right perpendicular and Delta = {(x(k), mu(k)) is an element of R x R : k = 0, 1,..., N} is a set of points. Suppose that Y-k is a random perturbation of mu(k) for k = 0, 1,..., N, and we set Delta* = {(x(k), Y-k) : k = 0, 1,..., N}. Let f(Delta) and f(Delta*) be linear fractal interpolation functions on I corresponding to the set of points Delta and Delta*, respectively. The value f(Delta*)(x) is random for all x is an element of I. In this paper, we show that the expectation of f(Delta*)(x) is f(Delta)(x). We also establish estimations for the variance of f(Delta*)(x) and the expectation of |f(Delta*)(x) - f(Delta)(x)|.

关键词： fractals interpolation fractal interpolation functions Random Data Sets

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Shape preserving constrained and monotonic rational quintic fractal interpolation functions

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INTERNATIONAL JOURNAL OF ADVANCES IN ENGINEERING SCIENCES AND APPLIED MATHEMATICS 2018年第1期10卷 15-33页

作者： Chand, A. K. B. Tyada, K. R. Indian Inst Technol Madras Dept Math Madras 600036 Tamil Nadu India

Shape preserving interpolants play important role in applied science and engineering. In this paper, we develop a new class of C-2-rational quintic fractal interpolation function (RQFIF) by using rational quintic functions of the form pi(t)/qi(t), where pi(t) is a quintic polynomial and qi(t) is a cubic polynomial with two shape parameters. The convergent result of the RQFIF to a data generating function in C-3 is presented. We derive simple restrictions on the scaling factors and shape parameters such that the developed rational quintic FIF lies above a straight line when the interpolation data with positive functional values satisfy the same constraint. Developing the relation between the attractors of equivalent dynamical systems, the constrained RQFIF can be extended to any general data. The positivity preserving RQFIF is a particular case of our result. In addition to this we also deduce the range on the IFS parameters to preserve the monotonicity aspect of given restricted type of monotonic data. The second derivative of the proposed RQFIF is irregular in a finite or dense subset of the interpolation interval, and matches with the second derivative of the classical rational quintic interpolation function whenever all scaling factors are zero. Thus, our scheme outperforms the corresponding classical counterpart, and the flexibility offered through the scaling factors is demonstrated through suitable examples.

关键词： fractal interpolation functions Rational quintic interpolation Constrained interpolation Dynamical systems Positivity Monotonicity

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fractal Jacobi Systems and Convergence of Fourier-Jacobi Expansions of fractal interpolation functions

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MEDITERRANEAN JOURNAL OF MATHEMATICS 2016年第6期13卷 3965-3984页

作者： Akhtar, Md. Nasim Prasad, M. Guru Prem Navascues, M. A. Indian Inst Technol Dept Math Gauhati 781039 Assam India Univ Zaragoza Dept Matemat Aplicada Zaragoza Spain

The fractal interpolation function (FIF) is a special type of continuous function on a compact subset of interpolating a given data set. They have been proved to be a very important tool in the study of irregular curves arising from financial series, electrocardiograms and bioelectric recording in general as an alternative to the classical methods. It is well known that Jacobi polynomials form an orthonormal system in with respect to the weight function , and . In this paper, a fractal Jacobi system which is fractal analogous of Jacobi polynomials is defined. The Weierstrass type theorem providing an approximation for square integrable function in terms of -fractal Jacobi sum is derived. A fractal basis for the space of weighted square integrable functions is found. The Fourier-Jacobi expansion corresponding to an affine FIF (AFIF) interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is established. The closeness of the original function to the Fourier-Jacobi expansion of the AFIF is proved for certain scale vector. Finally, the Fourier-Jacobi expansion corresponding to a non-affine smooth FIF interpolating certain data set is considered and its convergence in uniform norm and weighted-mean square norm is investigated as well.

关键词： fractal interpolation functions fractal Jacobi systems Schauder basis Fourier-Jacobi expansions

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Graph-Directed Coalescence Hidden Variable fractal interpolation functions

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Applied Mathematics 2016年第4期7卷 335-345页

作者： Md. Nasim Akhtar M. Guru Prem Prasad Department of Mathematics Indian Institute of Technology Guwahati Guwahati India

fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated Function System (IFS) corresponding to a data set is the graph of the FIF. Coalescence Hidden-variable fractal interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article, graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projection of the attractors on is the graph of the CHFIFs interpolating the corresponding data sets.

关键词： Iterated Function System Graph-Directed Iterated Function System fractal interpolation functions Coalescence Hidden Variable FIFs

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fractal interpolation functions with variable parameters and their analytical properties

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JOURNAL OF APPROXIMATION THEORY 2013年 175卷 1-18页

作者： Wang, Hong-Yong Yu, Jia-Shan Nanjing Univ Finance & Econ Sch Appl Math Nanjing 210023 Jiangsu Peoples R China

Based on a widely used class of iterated function systems (IFSs), a class of IFSs with variable parameters is introduced, which generates the fractal interpolation functions (FIFs) with more flexibility. Some analytical properties of these FIFs are investigated in the present paper. Their smoothness is first considered and the related results are presented in three different cases. The stability is then studied in the case of the interpolation points having small perturbations. Finally, the sensitivity analysis is carried out by providing an upper estimate of the errors caused by the slight perturbations of the IFSs generating these FIFs. (c) 2013 Elsevier Inc. All rights reserved.

关键词： Iterated function systems fractal interpolation functions Smoothness Stability Sensitivity

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Approximation with fractal radial basis functions

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2025年 454卷

作者： Kumar, D. Chand, A. K. B. Massopust, P. R. Indian Inst Technol Madras Dept Math Chennai 600036 India Tech Univ Munich TUM Dept Math D-85748 Munich Germany

The article reports on the construction of a general class of fractal radial basis functions (RBFs) in the literature. The fractal RBFs is defined through fractal perturbation of a RBF through suitable choice of iterated function system (IFS). A fractal RBF may be smooth depending on the choice of the germ function and the IFS parameters. Characterizations of conditionally strictly positive definite and strictly positive definite fractal functions are studied using the definition of k-times monotonicity. Furthermore, error estimates and shape-preserving properties for the approximants Pj j defined through linear combination of cardinal fractal RBFs are investigated. Several examples are presented to illustrate the convergence of the operator Pj j across various parameters, highlighting the advantages of the fractal approximant Pj j over the corresponding classical operator P . Finally, estimates for the box dimension of the graphs of approximants derived from fractal radial basis functions are given.

关键词： fractal interpolation functions Radial basis functions Strictly positive definite basis function Shape-preserving approximations Scattered interpolations Box dimension

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A comprehensive discussion on various methods of generating fractal-like Bézier curves

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第6期43卷 1-19页

作者： Vijay Kumar, Gurunathan Saravana Chand, A. K. B. Indian Inst Technol Madras Dept Math Chennai 600036 Tamil Nadu India Indian Inst Technol Guwahati Dept Math Gauhati 781039 Assam India Indian Inst Technol Madras Dept Engn Design Chennai 600036 Tamil Nadu India

This article explores various techniques for generating fractal-like B & eacute;zier curves in both 2D and 3D environments. It delves into methods such as subdivision schemes, Iterated Function System (IFS) theory, perturbation of B & eacute;zier curves, and perturbation of B & eacute;zier basis functions. The article outlines conditions on subdivision matrices necessary for convergence and demonstrates their use in creating an IFS with an attractor aligned to the convergent point of the subdivision scheme based on specified initial data. Additionally, it discusses conditions for obtaining a one-sided approximation of a given B & eacute;zier curve through perturbation. The article also addresses considerations for perturbed B & eacute;zier basis functions to construct fractal-like B & eacute;zier curves that remain within the convex hull polygon/polyhedron defined by control points. These methods find applications in various fields, including computer graphics, art, and design.

关键词： fractals Subdivision schemes Iterated function system B & eacute zier curves fractal interpolation functions

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fractal functions and fractal Dimensions on the Product of the Sierpiński Gaskets

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RESULTS IN MATHEMATICS 2025年第4期80卷 1-19页

作者： Jebali, Hajer Univ Monastir Fac Sci Monastir Dept Math Anal Probabil & Fractals Lab LR18ES17 Monastir 5000 Tunisia

In this paper, we construct a nonlinear fractal interpolation function on the product of the Sierpi & nacute;ski gaskets, and we prove that its graph is the unique attractor of an iterated function system defined, in a more general setting, by use of Rakotch contractions. Moreover, we use the H & ouml;lder continuity of pluriharmonic functions to estimate the fractal dimensions of the graph of the fractal interpolation function and those of the self-measure supported on it.

关键词： fractal interpolation functions fractal dimension Rakotch contractions Sierpi & nacute ski gasket Invariant measure Pluriharmonic functions

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fractal interpolation on the real projective plane

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NUMERICAL ALGORITHMS 2024年第2期96卷 557-582页

作者： Hossain, Alamgir Akhtar, Md. Nasim Navascues, Maria A. Presidency Univ Dept Math 86-1 Coll St Kolkata 700073 W Bengal India Univ Zaragoza Escuela Ingn & Arquitectura Dept Matemat Aplicada Zaragoza 500018 Spain

Formerly the geometry was based on shapes, but since the last centuries this founding mathematical science deals with transformations, projections, and mappings. Projective geometry identifies a line with a single point, like the perspective on the horizon line and, due to this fact, it requires a restructuring of the real mathematical and numerical analysis. In particular, the problem of interpolating data must be refocused. In this paper, we define a linear structure along with a metric on a projective space, and prove that the space thus constructed is complete. Then, we consider an iterated function system giving rise to a fractal interpolation function of a set of data.

关键词： Real projective plane fractal interpolation functions Real projective iterated function system Real projective fractal function

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