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Parameter Identification for Constrained Data Using a New Class of Rational fractal functions

NUMERICAL ANALYSIS AND APPLICATIONS 2021年第3期14卷 225-237页

作者： Katiyar, S. K. Chand, A. K. B. Jha, S. SRM Inst Sci & Technol Dept Math Chennai Tamil Nadu India Indian Inst Technol Madras Dept Math Chennai Tamil Nadu India Natl Inst Technol Dept Math Rourkela India

This paper sets a theoretical foundation for applications of fractal interpolation functions (FIFs). We construct rational cubic spline FIFs (RCSFIFs) with a quadratic denominator involving two shape parameters. The elements of the iterated function system (IFS) in each subinterval are identified so that the graph of the resulting C-1-RCSFIF lies within a prescribed rectangle. These parameters include, in particular, conditions on the positivity of the C-1-RCSFIF. The problem of visualization of constrained data is also addressed when a data set is lying above a straight line and the proposed fractal curve is required to lie on the same side of the line. We illustrate our interpolation scheme with some numerical examples.

关键词： iterated function system fractal interpolation functions rational cubic fractal functions rational cubic interpolation constrained interpolation positivity

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CONSTRUCTION OF fractal SURFACES

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2020年第2期28卷

作者： Navascues, M. A. Mohapatra, R. N. Akittar, M. N. Univ Zaragoza Escuela Ingn & Arguitectura C Maria de Luna 3 Zaragoza 50018 Spain Univ Cent Florida Dept Math 4393 Andromeda Loop Orlando FL 32816 USA Aliah Univ Dept Math & Stat Kolkata 700156 India

The paper approaches the construction of fractal surfaces of interpolation and approximation on the basis of a fractal perturbation of any mapping defined on a rectangle. Conditions for the differentiability of these elements are also provided. The fractal surfaces obtained may be used for the approximation of real-world data. The method proposed does not require any restriction on the type of data. Furthermore, the present approach does not imply the solution of large systems of equations. The paper considers both the continuous and the discontinuous case.

关键词： fractal interpolation functions Iterated Function Systems fractals Surfaces

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MULTIVARIATE AFFINE fractal interpolation

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2020年第7期28卷

作者： Navascues, M. A. Katiyar, S. K. Chand, A. K. B. Univ Zaragoza Dept Matemat Aplicada C Maria de Luna 3 Zaragoza 50018 Spain SRM Inst Sci & Technol Dept Math Chennai 603203 Tamil Nadu India Indian Inst Technol Madras Dept Math Chennai 600036 Tamil Nadu India

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 from real world signals, which are usually difficult to represent using the classical approaches. The affine fractal interpolants constitute a generalization of the broken line interpolation, which appears as a particular case of the linear self-affine functions for specific values of the scale parameters. We study the Lp convergence of this type of interpolants for 1 <= p < extending in this way the results available in the literature. In the second part, the affine approximants are defined in higher dimensions via product of interpolation spaces, considering rectangular grids in the product intervals. The associate operator of projection is considered. Some properties of the new functions are established and the aforementioned operator on the space of continuous functions defined on a multidimensional compact rectangle is studied.

关键词： Iterated Function System fractals fractal interpolation functions Smooth fractal Function fractal Operator

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On the distributions of fractal functions that interpolate data points with Gaussian noise

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CHAOS SOLITONS & fractalS 2020年 135卷 109743-109743页

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

We first construct a fractal interpolation function f() corresponding to the data set Delta(mu) = {(x(k), mu(k)) : k = 0, 1,, N}, where x(0) ) corresponding to the set Delta(Y) = {(x(k), Y-k) : k = 0,1,...,N}. All val... 详细信息

We first construct a fractal interpolation function f(< T mu >) corresponding to the data set Delta(mu) = {(x(k), mu(k)) : k = 0, 1,, N}, where x(0) < x(1) < x(2) < ... < x(N). Let Y-k = mu(k) + epsilon(k) for k = 0, 1,..., N, where epsilon(k) is a stochastic disturbance term with zero expectation and finite variance. Then we establish a fractal interpolation function f(< T mu >) corresponding to the set Delta(Y) = {(x(k), Y-k) : k = 0,1,...,N}. All values f(< TY >) (x) are random. For x is an element of [x(0), x(N)], we show that the expectation of f(< TY >)(x) is equal to f(< T mu >) (x) and we also give an estimation for the variance of f(< TY >) (x). If epsilon(0), epsilon(1),..., epsilon(N) are independent Gaussian random variables, then we show that f(< TY >) (x) is also a Gaussian random variable for each x subset of [x(0), x(N)]. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： fractals interpolation fractal interpolation functions Gaussian noise

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Approximation by shape preserving fractal functions with variable scalings

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CALCOLO 2021年第1期58卷 8-8页

作者： Jha, Sangita Chand, A. K. B. Navascues, M. A. Natl Inst Technol Rourkela Dept Math Rourkela 769008 India Indian Inst Technol Madras Dept Math Chennai 600036 Tamil Nadu India Univ Zaragoza Escuela Ingn & Arquitectura Dept Matemat Aplicada Zaragoza 500018 Spain

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions f alpha is an element of C(I), where alpha is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the alpha-fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the alpha-fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the alpha-fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Muntz theorem in C[0,1].

关键词： fractals fractal interpolation functions Smoothing Shape preservation Chebyshev system Muntz approximation

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Image compression using fractal multiwavelet transform

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JOURNAL OF ANALYSIS 2020年第3期28卷 769-789页

作者： Akhtar, Md Nasim Prasad, M. Guru Prem Kapoor, G. P. Indian Inst Technol Guwahati Dept Math Gauhati 781039 Assam India Indian Inst Technol Kanpur Dept Math & Stat Kanpur Uttar Pradesh India

Multiwavelets occur in wavelet theory for more general multiresolution analysis (MRA) using several scaling functions. The muliwavelets feature orthogonality, short support, symmetry, approximation order and regularity at a time which is not possible by using only a single wavelet system. In the present work, theory of MRA and wavelet using several scaling functions is reviewed to define Discrete Multiwavelet Transform and fractal Multiwavelet Transform in matrix form, using a multifilter. As an application the defined transforms are applied to an image for its compression and the resulting entropy and PSNR values are compared with those occurring with earlier used transforms like Haar and D4 transforms.

关键词： fractal interpolation functions Multiresolution analysis Multiwavelets Image compression 28A80 42C40 94A08

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Dimensional Analysis of α-fractal functions

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RESULTS IN MATHEMATICS 2021年第4期76卷 186-186页

作者： Jha, S. Verma, S. NIT Rourkela Dept Math Rourkela 769008 India IIIT Allahabad Dept Appl Sci Allahabad 211015 Uttar Pradesh India

We provide a rigorous study on dimensions of fractal interpolation functions defined on a closed and bounded interval of R which are associated to a continuous function with respect to a base function, scaling functions and a partition of the interval. In particular, we calculate an exact estimation of box dimension of alpha-fractal functions under suitable hypotheses on the iterated function system.

关键词： Iterated function systems fractal interpolation functions Hausdorff dimension box dimension open set condition

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A new class of rational cubic spline fractal interpolation function and its constrained aspects

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APPLIED MATHEMATICS AND COMPUTATION 2019年 346卷 319-335页

作者： Katiyar, S. K. Chand, A. K. B. Kumar, G. Saravana Indian Inst Technol Madras Dept Math Madras 600036 Tamil Nadu India Indian Inst Technol Madras Dept Engn Design Madras 600036 Tamil Nadu India SRM Inst Sci & Technol Dept Math Madras 603203 Tamil Nadu India

This paper pertains to the area of shape preservation and sets a theoretical foundation for the applications of preserving constrained nature of a given constraining data in fractal interpolation functions (FIFs) techniques. We construct a new class of rational cubic spline FIFs (RCSFIFs) with a preassigned quadratic denominator with two shape parameters, which includes classical rational cubic interpolant [Appl. Math. Comp., 216 (2010), pp. 2036-2049] as special case and improves the sufficient conditions for positivity. Convergence analysis of RCSFIF to the original function in C-1 is studied. In order to meet the needs of practical design or overcome the drawback of the tension effect in the proposed RCSFIFs, we improve our method by introducing a new tension parameter w, and construct a new class of rational cubic spline FIFs with three shape parameters. The scaling factors and shape parameters have a predictable adjusting role on the shape of curves. The elements of the rational iterated function system in each subinterval are identified befittingly so that the graph of the resulting C-1-rational cubic spline FIF constrained (i) within a prescribed rectangle (ii) above a prescribed straight line (iii) between two piecewise straight lines. These parameters include, in particular, conditions on the positivity of the C-1-rational cubic spline FIF. Several numerical examples are presented to ascertain the correctness and usability of developed scheme and to suggest how these schemes outperform their classical counterparts. (C) 2018 Elsevier Inc. All rights reserved.

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

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SHAPE PRESERVING RATIONAL QUARTIC fractal functions

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2019年第8期27卷

作者： Katiyar, S. K. Chand, A. K. B. SRM Inst Sci & Technol Dept Math Chennai 603203 Tamil Nadu India Indian Inst Technol Madras Chennai 600036 Tamil Nadu India

The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct alpha-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the alpha-fractal rational quartic spline when the original function is in C-4(J). By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the alpha-fractal rational quartic spline to C-2. The elements of the iterated function system are identified befittingly so that the class of alpha-fractal function Q(alpha) incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ Q. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

关键词： fractals Iterated Function System fractal interpolation functions Positivity Monotonicity Convexity

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fractal CONVOLUTION: A NEW OPERATION BETWEEN functions

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FRACTIONAL CALCULUS AND APPLIED ANALYSIS 2019年第3期22卷 619-643页

作者： Navascues, Maria A. Massopust, Peter R. Univ Zaragoza Dept Matemat Aplicada Calle Maria de Luna 3 Zaragoza 50018 Spain Tech Univ Munich Ctr Math Res Unit M15 Boltzmannstr 3 D-85748 Munich B Germany

In this paper, we define an internal binary operation between functions called fractal convolution that when applied to a pair of mappings generates a fractal function. This is done by means of a suitably defined iterated function system. We study in detail this operation in L-p spaces and in sets of continuous functions in a way that is different from the previous work of the authors. We develop some properties of the operation and its associated sets. The lateral convolutions with the null function provide linear operators whose characteristics are explored. The last part of the article deals with the construction of convolved fractals bases and frames in Banach and Hilbert spaces of functions.

关键词： fractals iterated function systems fractal interpolation functions function spaces

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