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Some measure-theoretic aspects of fractal functions: invariant measures and function spaces

QUAESTIONES MATHEMATICAE 2024年第10期47卷 2069-2093页

作者： Pandey, Megha Som, Tanmoy Verma, Saurabh IIT BHU Dept Math Sci Varanasi 221005 India IIIT Allahabad Dept Appl Sci Allahabad Prayagraj 211015 India

Barnsley (1986) introduced the concept of fractal interpolation functions (FIFs). Thereafter, numerous theories have been developed concerning FIFs and their properties. In this article, we explore some measure theoretic aspects of FIFs generated by the method given by Barnsley. We discuss some properties of the invariant measures supported on the graph of FIF and compare our results with the previously developed theories. In the sequel, we study fractal transformation between two FIFs satisfying the same data of interpolation. Later, we define some function spaces and present a few results that give conditions under which the FIF becomes an element of those spaces. With the help of these spaces, we estimate the fractal dimension of FIFs. We also give several remarks, notes, and examples in support of our study.

关键词： fractal interpolation functions iterated function system invariant measure fractal transformation L-p-spaces convex-Lipschitz space dimensional space

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Collage theorems, invertibility and fractal functions

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FRACTIONAL CALCULUS AND APPLIED ANALYSIS 2024年第3期27卷 1112-1135页

作者： Navascues, Maria A. Mohapatra, Ram N. Univ Zaragoza Dept Math Calle Maria de Luna 3 Zaragoza 50018 Spain Univ Cent Florida Orlando FL 32816 USA

Collage Theorem provides a bound for the distance between an element of a given space and a fixed point of a self-map on that space, in terms of the distance between the point and its image. We give in this paper some results of Collage type for Reich mutual contractions in b-metric and strong b-metric spaces. We give upper and lower bounds for this distance, in terms of the constants of the inequality involved in the definition of the contractivity. Reich maps contain the classical Banach contractions as particular cases, as well as the maps of Kannan type, and the results obtained are very general. The middle part of the article is devoted to the invertibility of linear operators. In particular we provide criteria for invertibility of operators acting on quasi-normed spaces. Our aim is the extension of the Casazza-Christensen type conditions for the existence of inverse of a linear map defined on a quasi-Banach space, using different procedures. The results involve either a single map or two operators. The latter case provides a link between the properties of both mappings. The last part of the article is devoted to study the construction of fractal curves in Bochner spaces, initiated by the first author in a previous paper. The objective is the definition of fractal curves valued on Banach spaces and Banach algebras. We provide further results on the fractal convolution of operators, defined in the same reference, considering in this case the nonlinear operators. We prove that some properties of the initial maps are inherited by their convolutions, if some conditions on the elements of the associated iterated function system are satisfied. In the last section of the paper we use the invertibility criteria given before in order to obtain perturbed fractal spanning systems for quasi-normed Bochner spaces composed of Banach-valued integrable maps. These results can be applied to Lebesgue spaces of real valued functions as a particular case.

关键词： fractal functions (primary) Collage theorem invertibility of linear operators fractal interpolation functions fractals

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Event-Triggered Impulsive Controller Design for Synchronization of Delayed Chaotic Neural Networks and Its fractal Reconstruction: An Application to Image Encryption

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APPLIED MATHEMATICAL MODELLING 2023年 115卷 490-512页

作者： Mohanrasu, S. S. Udhayakumar, K. Priyanka, T. M. C. Gowrisankar, A. Banerjee, Santo Rakkiyappan, R. Bharathiar Univ Dept Math Coimbatore 641046 Tamil Nadu India UAE Univ Coll Sci Dept Math Sci Al Ain U Arab Emirates Vellore Inst Technol Sch Adv Sci Dept Math Vellore 632014 Tamil Nadu India Politecn Torino Dipartimento Sci Matematiche Corso Duca Abruzzi 24 I-10129 Turin Italy

This paper examines synchronization of delayed chaotic neural networks with impulsive control, event-triggered impulsive control, and event-triggered delayed impulsive control. Lyapunov-based event-triggered mechanism is constructed to derive few sufficient condi-tions in terms of Linear Matrix Inequalities (LMIs). The feedback and event-triggered gain matrices are evaluated with the help of numerical packages. The fractal interpolation func-tions are employed to reconstruct the chaotic sequence of the neural networks since the chaotic attractor has self-similar properties and coincides with the fractal functions. In this study, a linear fractal interpolation is used to reconstruct both the master-slave chaotic at -tractor and the encrypted chaotic attractor, with optimized scaling ***, an application of the image encryption algorithm is given in terms of RSA public -key cryptosystems, permutation, and finite field diffusion for the chaotic sequence gener-ated by fractal interpolation functions.(c) 2022 Elsevier Inc. All rights reserved.

关键词： Chaotic neural networks fractal interpolation functions Linear matrix inequalities Vertical scaling factors RSA public -key cryptosystems Image encryption

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New Type of fractal functions for the General Data Sets

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ACTA APPLICANDAE MATHEMATICAE 2023年第1期187卷 1-23页

作者： Verma, Manuj Priyadarshi, Amit Indian Inst Technol Delhi Dept Math New Delhi 110016 India

In this paper, we prove the existence of the fractal interpolation function corresponding to a general data set using the Rakotch contraction theory and iterated function system. We also prove the existence of the fractal measure supported on the graph of the fractal interpolation function. We emphasize the fact that our theory covers the fractal interpolation theory for finite cases, countably infinite cases, and many more. We establish dimensional results for the graph of the fractal interpolation function for the general data sets.

关键词： Iterated function systems fractal interpolation functions Hausdorff dimension Box dimension Rakotch contraction

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Vector-valued fractal functions: fractal dimension and fractional calculus

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INDAGATIONES MATHEMATICAE-NEW SERIES 2023年第4期34卷 830-853页

作者： Verma, Manuj Priyadarshi, Amit Verma, Saurabh Indian Inst Technol Delhi Dept Math New Delhi 110016 India Indian Inst Informat Technol Allahabad Dept Appl Sci Prayagraj 211015 India

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann-Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann-Liouville fractional integral of a vector-valued fractal interpolation function.(c) 2023 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

关键词： Iterated function systems fractal interpolation functions Hausdorff dimension Box dimension Open set condition Riemann-Liouville fractional integral

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Dimensions of new fractal functions and associated measures

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NUMERICAL ALGORITHMS 2023年第2期94卷 817-846页

作者： Verma, Manuj Priyadarshi, Amit Indian Inst Technol Delhi Dept Math New Delhi 110016 India

In this paper, we obtain a vector-valued fractal interpolation function in a more general setting by using the Rakotch fixed point theory and the iterated function system. We also show the existence of the Borel probability measure, widely known as a fractal measure, supported on the graph of this vector-valued fractal interpolation function. We obtain the Hausdorff dimension and the box-counting dimension of the graph of this more general fractal function using some function spaces. We obtain bounds for the fractal dimension of the graph of the Riemann-Liouville fractional integral of this fractal function. Using our techniques, we also calculate the fractal dimensions of the graphs of some fractal interpolation functions.

关键词： Iterated function systems fractal interpolation functions Hausdorff dimension Box dimension Rakotch contraction

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L^p-Approximation Using fractal functions on the Sierpiriski Gasket

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RESULTS IN MATHEMATICS 2022年第2期77卷 1-17页

作者： Agrawal, Vishal Som, Tanmoy Indian Inst Technol BHU Dept Math Sci Varanasi 221005 Uttar Pradesh India

We are concerned with the approximation of functions by fractal functions with respect to L-p-norm on the Sierphiski gasket. We define the alpha-fractal function in L-p space. The properties such as topological isomorphism and many others, which are closely associated with the fractal operator will be discussed in more detail. We also prove the existence of a non-trivial closed invariant subspace for the fractal operator. Additionally, we define set-valued mapping and discuss some useful properties.

关键词： fractal interpolation functions fractal dimension Sierphiski gasket self-similar measure L-p-approximation

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ON fractal DIMENSIONS OF fractal functions USING FUNCTION SPACES

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BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY 2022年第3期106卷 470-480页

作者： Chandra, Subhash Bbas, Syed A. Indian Inst Technol Mandi Sch Basic Sci Kamand 175005 Himachal Prades India

Based on the work of Mauldin and Williams ['On the Hausdorff dimension of some graphs', Trans. Amer. Math. Soc. 298(2) (1986), 793-803] on convex Lipschitz functions, we prove that fractal interpolation functions belong to the space of convex Lipschitz functions under certain conditions. Using this, we obtain some dimension results for fractal functions. We also give some bounds on the fractal dimension of fractal functions with the help of oscillation spaces.

关键词： fractal interpolation functions convex-Lipschitz space oscillation space Hausdorff dimension box dimension

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fractal SURFACES INVOLVING RAKOTCH CONTRACTION FOR COUNTABLE DATA SETS

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2024年第2期32卷 2440002-2440002页

作者： Verma, Manuj Priyadarshi, Amit Indian Inst Technol Delhi Dept Math New Delhi India

In this paper, we prove the existence of the bivariate fractal interpolation function using the Rakotch contraction theory and iterated function system for a countable data set. We also give the existence of the invariant Borel probability measure supported on the graph of the bivariate fractal interpolation function. In particular, we highlight that our theory encompasses the bivariate fractal interpolation theory in both finite and countably infinite settings available in literature.

关键词： Iterated Function Systems fractal interpolation functions Hausdorff Dimension Box Dimension Rakotch Contraction

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SOME PROPERTIES OF THE fractal CONVOLUTION OF functions

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FRACTIONAL CALCULUS AND APPLIED ANALYSIS 2021年第6期24卷 1735-1757页

作者： Navascues, Maria A. Mohapatra, Ram N. Chand, Arya K. B. Univ Zaragoza Dept Matemat Aplicada Escuela Ingn & Arquitectura Zaragoza Spain Univ Cent Florida Dept Math Orlando FL 32816 USA Indian Inst Technol Madras Dept Math Madras Tamil Nadu India

We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fre acute accent chet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.

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

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