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Toward a Unified Methodology for fractal Extension of Various Shape Preserving Spline Interpolants 1

2nd International Conference on Mathematics and Computing (ICMC)

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

ISBN: (数字)9788132224525

ISBN: (纸本)9788132224525;9788132224518

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 self-referential set. The novelty of fractal interpolants lies in their ability to model a data set with either a smooth or a nonsmooth function depending on the problem at hand. To broaden their horizons, some special class of fractal interpolants are introduced and their shape preserving aspects are investigated recently in the literature. In the current article, we provide a unified approach for the fractal generalization of various traditional nonrecursive polynomial and rational splines. To this end, first we shall view polynomial/rational FIFs as alpha-fractal functions corresponding to the traditional nonrecursive splines. The elements of the iterated function system are identified befittingly so that the class of alpha-fractal function f(alpha) incorporates the geometric features such as positivity, monotonicity, and convexity in addition to the regularity inherent in the generating function f. This general theory in conjuction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions. Even though the results obtained in this article are generally enough, we wish to apply it on a specific rational cubic spline with two free shape parameters.

关键词： fractals Iterated function system fractal interpolation functions Rational cubic fractal functions Rational cubic interpolation Positivity

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Construction of orthogonal multi-wavelets using generalized-affine fractal interpolation functions

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IMA JOURNAL OF APPLIED MATHEMATICS 2009年第6期74卷 904-933页

作者： Bouboulis, P. Univ Athens Dept Informat & Telecommun Telecommun & Signal Pr Athens 15784 Greece

We present a new construction of fractal interpolation surfaces defined on arbitrary rectangular lattices. We use this construction to form finite sets of fractal interpolation functions (FIFs) that generate multiresolution analyses of L(2)(R(2)) of multiplicity r. These multiresolution analyses are based on the dilation properties of the construction. The associated multi-wavelets are orthogonal and discontinuous functions. We give concrete examples to illustrate the method and generalize it to form multiresolution analyses of L(2)(R(d)), d > 2. To this end, we prove some results concerning the Holder exponent of FIFs defined on [0, 1](d).

关键词： fractal interpolation functions fractal interpolation surfaces fractals moments Holder multi-wavelets

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Box dimension and fractional integral of linear fractal interpolation functions

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JOURNAL OF APPROXIMATION THEORY 2009年第1期161卷 187-197页

作者： Ruan, Huo-Jun Su, Wei-Yi Yao, Kui Zhejiang Univ Dept Math Hangzhou 310027 Zhejiang Peoples R China Nanjing Univ Dept Math Nanjing 210093 Peoples R China Nanjing Normal Univ Dept Math Nanjing 210097 Peoples R China

In this paper, we present a new method to calculate the box dimension of a graph of continuous functions. Using this method, we obtain the box dimension formula for linear fractal interpolation functions (FIFs). Furthermore we prove that the fractional integral of a linear FIF is also a linear FIF and in some cases, there exists a linear relationship between the order of fractional integral and box dimension of two linear FIFs. (C) 2008 Elsevier Inc. All rights reserved.

关键词： fractal interpolation functions Box dimension Fractional integral

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AFFINE fractal functions AS BASES OF CONTINUOUS functions

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QUAESTIONES MATHEMATICAE 2014年第3期37卷 415-428页

作者： Navascues, M. A. Univ Zaragoza Dept Matemat Aplicada Escuela Ingn & Arquitectura Zaragoza 50018 Spain

The objective of the present paper is the study of affine transformations of the plane, which provide self-affine curves as attractors. The properties of these curves depend decisively of the coefficients of the system of affnities involved. The corresponding functions are continuous on a compact interval. If the scale factors are properly chosen one can define Schauder bases of C[a, b] composed by affine fractal functions close to polygonals. They can be chosen bounded. The basis constants and the biorthogonal sequence of coefficient functionals are studied.

关键词： 28A80 58C05 65D05 65D10 26A18 fractal interpolation functions iterated function systems Schauder bases

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Shape preservation of scientific data through rational fractal splines

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CALCOLO 2014年第2期51卷 329-362页

作者： Chand, A. K. B. Vijender, N. Navascues, M. A. Indian Inst Technol Dept Math Chennai 600036 Tamil Nadu India Univ Zaragoza Dept Matemat Aplicada Escuela Ingn & Arquitectura Zaragoza 50018 Spain

fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form where and are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.

关键词： fractals Iterated function systems fractal interpolation functions Rational cubic fractal functions Rational cubic interpolation Monotonicity Positivity Convexity

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BIORTHOGONAL MULTIWAVELETS RELATED BY DIFFERENTIATION

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INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING 2014年第2期12卷 1450021-1450021页

作者： Phan Nguyen Loyola Univ Dept Math Sci New Orleans LA 70118 USA

We provide a procedure for constructing biorthogonal multiwavelets from a family of biorthogonal multiscaling functions compactly supported on [-1, 1]. The scaling vectors and the associated multiwavelets are piecewise continuously differentiable, symmetrical and possess approximation order three. The construction of scaling vectors is accomplished using quadratic fractal interpolation functions. The filters corresponding to scaling vectors possess certain properties which enable us to construct a new pair of biorthogonal scaling vectors and associated multiwavelets with different regularity and approximation order, related to the old ones by differentiation. The old and new biorthogonal multiwavelet systems give rise to compactly supported biorthogonal multiwavelet basis for the space of divergence-free vector fields on the upper half plane with the Navier boundary condition.

关键词： Biorthogonal multiwavelets fractal interpolation functions Strela's two-scale transform

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Dependence of friction coefficient on the resolution and fractal dimension of metallic fracture surfaces

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INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES 2013年第20-21期50卷 3106-3118页

作者： Panagouli, Olympia K. Iordanidou, Kyriaki Univ Thessaly Dept Civil Engn Lab Struct Anal & Design Volos 38334 Volos Greece

The paper studies the friction mechanism that is developed between metallic fracture interfaces of fractal geometry. The main purpose of the paper is to investigate how both the resolution and the fractal dimension of the interface affect the friction mechanism. Friction is assumed to be the result of the involving phenomena and the gradual plastifications of the fractal interface asperities. For that, a fractured body with an elastic-plastic behaviour is assumed, in which the fracture interface has fractal geometry. Three different cases are considered, corresponding to fractal interfaces f((m)) with different fractal dimensions. For each fractal interface, a number of classical problems is considered, corresponding to different values of the resolution delta(n). In the limit of the finest resolution useful conclusions are derived for the friction coefficient for the scale ranges studied here. Finally, the influence of the applied normal loading to the friction mechanism is investigated with respect to the fractal dimension and the fractal resolution of the interfaces. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Roughness fractal interpolation functions Resolution Friction coefficient Size effects

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Numerical integration of affine fractal functions

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2013年 252卷 169-176页

作者： Navascues, M. A. Sebastian, M. V. Univ Zaragoza Escuela Ingn & Arquitectura Dept Matemat Aplicada Zaragoza 50018 Spain Acad Gen Mil Zaragoza Ctr Univ Def Zaragoza 50090 Spain

This paper studies a method for the numerical integration and representation of functions defined through their samples, when the original "signal" is not explicitly known, but it shows experimentally some kind of self-similarity. In particular, we propose a methodology based on fractal interpolation functions for the computation of the integral that generalize the compound trapezoidal rule. The convergence of the procedure is proved with the only hypothesis of continuity. The rate of convergence is specified in the case of original Holder-continuous functions, but not necessarily smooth. (c) 2012 Elsevier B.V. All rights reserved.

关键词： fractal interpolation functions Numerical integration Quadrature Affinities fractals

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fractal interpolation surfaces derived from fractal interpolation functions

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2007年第2期336卷 919-936页

作者： Bouboulis, P. Dalla, L. Univ Athens Dept Informat Telecommun Telecommun & Signal P GR-15784 Athens Greece Univ Athens Dept Math Mathemat Anal GR-15784 Athens Greece

Based on the construction of fractal interpolation functions, a new construction of fractal interpolation Surfaces on arbitrary data is presented and some interesting properties of them are proved. Finally, a lower bo... 详细信息

关键词： fractal interpolation functions fractal interpolation surfaces box counting dimension fractals smooth fractal surfaces

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fractal BASES OF Lp SPACES

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2012年第2期20卷 141-148页

作者： Navascues, M. A. Univ Zaragoza Dept Matemat Aplicada Escuela Ingn & Aquitectura Ctr Politecn Super Ingenieros Zaragoza 50018 Spain

The methodology of fractal sets generates new procedures for the analysis of functions whose graphs have a complex geometric structure. In the present paper, a method for the definition of fractal functions is described. The new mappings are perturbed versions of classical bases as Legendre polynomials, etc. The new elements are non-differentiable and may serve as models for pseudo-random behaviour. The proposed fractal functions have good algebraic properties and good approximation properties as well. In the present paper it is proved that they constitute bases for the most important functional spaces as, for instance, the Lebesgue spaces L-p(I) (1 <= p < infinity), where I is a compact interval in the reals.

关键词： fractal interpolation functions Least Squares Approximation Orthogonal Systems Bases of Functional Spaces fractals

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