Box dimension and fractional integral of linear fractal interpolation functions

Ruan, Huo-Jun; Su, Weiyi; Yao, Kui

doi:10.1016/j.jat.2008.08.012

Cited by 113 publications

(32 citation statements)

References 8 publications

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“…Then, a fractal interpolation function g(x) defined as above is called a linear fractal interpolation function. In [15], the authors chose…”

Section: Proposition 23mentioning

confidence: 99%

“…Riemann-Liouville fractional integral of any continuous functions which have finite points of unbounded variation still are 1-dimensional [9]. There exist continuous functions whose box dimension are bigger than 1, such as Weierstrass function [1], Besicovitch function [17], linear fractal interpolation functions [15]. More details about functions of unbounded variation and definition of fractal functions can be found in [6].…”

Section: Introductionmentioning

confidence: 99%

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Fractal dimensions of fractional integral of continuous functions

Liang

2016

Acta. Math. Sin.-English Ser.

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In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann-Liouville integral of a continuous function f (x) of order v(v > 0) which is written as D −v f (x) has been proved to still be continuous and bounded. Furthermore,With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann-Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f (x) satisfying Hölder condition, upper box dimension of Riemann-Liouville fractional integral of f (x) seems no more than upper box dimension of f (x). Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann-Liouville integral of a continuous function satisfying Hölder condition of order v(v > 0) is strictly less than 2 − v. Riemann-Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl-Marchaud fractional derivative of certain continuous functions have been estimated.

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“…Then, a fractal interpolation function g(x) defined as above is called a linear fractal interpolation function. In [15], the authors chose…”

Section: Proposition 23mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractal dimensions of fractional integral of continuous functions

Liang

2016

Acta. Math. Sin.-English Ser.

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“…They have been developed both in theory and applications by many authors; see for example [25,40,89,90,95,101,108,109,120]. They provide an alternative view on wavelets, [40,53].…”

Section: Fractal Continuationmentioning

confidence: 99%

Developments in fractal geometry

Barnsley¹,

Vince²

2013

Bull. Math. Sci.

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Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor-repeller pairs and the Conley "landscape picture" for an IFS.

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“…In recent years, there has been much discussion on fractal dimension of fractional calculus of continuous functions, such as Weierstrass function, [1][2][3] Besicovitch function, 4,5 fractal interpolation functions 6,7 and other special continuous functions. [8][9][10][11] Fractional calculus such as RiemannLiouville fractional calculus 4,12 which has been used to investigate fractal curves is an important tool in the fractal analysis.…”

Section: Introductionmentioning

confidence: 99%