On a class of fractal functions with graph Hausdorff dimension 2

Xie, Tingfan; Zhou, S. P.

doi:10.1016/j.chaos.2005.12.038

Cited by 27 publications

(6 citation statements)

References 11 publications

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“…Barnsley made research on the linear fractal interpolation function in [8]. The examples of continuous functions with the Box dimension two, which are differentiable nowhere on I, can be found in [9,10].…”

Section: Example 1 ([13]) the Weierstrass Functionmentioning

confidence: 99%

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Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions

Liang

2022

Mathematics

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In the present paper, we try to estimate the fractal dimensions of the linear combination of continuous functions with different fractal dimensions. Initially, a general method to calculate the lower and the upper Box dimension of the sum of two continuous functions by classifying all the subsequences into different sets has been proposed. Further, we discuss the majority of possible cases of the sum of two continuous functions with different fractal dimensions and obtain their corresponding fractal dimensions estimation by using that general method. We prove that the linear combination of continuous functions having no Box dimension cannot keep the fractal dimensions closed. In this way, we have figured out how the fractal dimensions of the linear combination of continuous functions change with certain fractal dimensions.

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Section: Example 1 ([13]) the Weierstrass Functionmentioning

confidence: 99%

“…For example, the Weierstrass function W(x) ∈ F 2−α . The functions constructed in [9,10] belong to F 2 . In [19], we can find a special fractal function belonging to F 1 .…”

Section: The Linear Combination Of Two Functions With Different Fract...mentioning

confidence: 99%

Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions

Liang

2022

Mathematics

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“…An examples of continuous functions belong to 2 D I has been given in Ref. [23] or see following Example 3.6. Thus we know 1 D I and 2 D I are both not empty.…”

Section: Example 34[6] Weierstrass Function W (X)mentioning

confidence: 99%

“…(1) self-affine curves such as [2,6,21]. (2) Particular functions like Weierstrass and their variant types (some called Besicovitch or Besicovitch type functions) such as [1,3,8,17,18,23]. (3) Fractal integral or derivative on these particular functions such as [7,9,13].…”

Section: Introductionmentioning

confidence: 99%

On Fractal Features and Fractal Linear Space About Fractal Continuous Functions

Xiao¹

2021

Preprint

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This paper investigates fractal dimension of linear combination of fractal continuous functions with the same or different fractal dimensions. It has been proved that: (1) BV I all fractal continuous functions with bounded variation is fractal linear space; (2) 1 D I all fractal continuous functions with Box dimension one is a fractal linear space; (3) s D I all fractal continuous functions with identical Box dimension s(1 < s ≤ 2) is surprisingly a non-fractal linear space, even non-fractal linear manifold, beyond our initial expectation, because the Box dimension of linear combination of fractal continuous functions can take any real number in [1, s) if it exists, and some different upper and lower Box dimension if it does not exit. This attracts our interests to probe into fractal characteristics of s D I , and get some suggesting results.

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“…Barnsley and Ruan have made research on the linear fractal interpolation functions in [5,6], respectively. Moreover, there exist certain particular examples of one-dimensional fractal functions discussed in [7][8][9][10][11][12][13][14] and two-dimensional fractal functions constructed in [15,16]. For Hölder continuous functions, ref.…”

Section: Introductionmentioning

confidence: 99%

On the Lower and Upper Box Dimensions of the Sum of Two Fractal Functions

Liang

2022

Fractal Fract

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Let f and g be two continuous functions. In the present paper, we put forward a method to calculate the lower and upper Box dimensions of the graph of f+g by classifying all the subsequences tending to zero into different sets. Using this method, we explore the lower and upper Box dimensions of the graph of f+g when the Box dimension of the graph of g is between the lower and upper Box dimensions of the graph of f. In this case, we prove that the upper Box dimension of the graph of f+g is just equal to the upper Box dimension of the graph of f. We also prove that the lower Box dimension of the graph of f+g could be an arbitrary number belonging to a certain interval. In addition, some other cases when the Box dimension of the graph of g is equal to the lower or upper Box dimensions of the graph of f have also been studied.

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On a class of fractal functions with graph Hausdorff dimension 2

Cited by 27 publications

References 11 publications

Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions

Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions

On Fractal Features and Fractal Linear Space About Fractal Continuous Functions

On the Lower and Upper Box Dimensions of the Sum of Two Fractal Functions

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