On the box-counting dimension of graphs of harmonic functions on the Sierpiński gasket

Sahu, Abhilash; Priyadarshi, Amit

doi:10.1016/j.jmaa.2020.124036

Cited by 41 publications

(12 citation statements)

References 31 publications

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“…Moreover, he proved the same type result of Mauldin and Williams for the lower box dimension. Bayart and Heurteaux [5] also proved similar result for Hausdorff dimension β = 2, and raised the question for β ∈ [1,2]. Recently in 2013, Jia Liu and Jun Wu [7] solved the question which was raised by Bayart and Haurteaux.…”

Section: Introductionmentioning

confidence: 76%

“…Later Hardin and Massopust [4] constructed fractal interpolation functions from R n to R m and also given the formula for calculating the box dimension. We encourage the reader to see some recent works on fractal dimension of fractal functions defined on different domains such as Sierpinski gasket [1,25], rectangular domain [21] and interval [23,24]. To the best of our knowledge, we may say that there is no work available for dimension of complex-valued fractal functions.…”

Section: Introductionmentioning

confidence: 99%

“…Recently in 2013, Jia Liu and Jun Wu [7] solved the question which was raised by Bayart and Haurteaux. More preciesly they proved that, for any β ∈ [1,2], each continuous function on [0, 1] can be decomposed into sum of two continuous functions, where each have Hausdorff dimension β. Falconer and Fraser [9] found an upper bound for upper box dimension of graph of sum of two continuous functions which depends on dimension of both of graphs.…”

Section: Introductionmentioning

confidence: 99%

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Fractal dimension for a class of complex-valued fractal interpolation functions

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There are many research papers dealing with fractal dimension of real-valued fractal functions in the recent literature. The main focus of the present paper is to study fractal dimension of complex-valued functions. This paper also highlights the difference between dimensional results of the complexvalued and real-valued fractal functions. In this paper, we study the fractal dimension of the graph of complex-valued function g(x) + ih(x), compare its fractal dimension with the graphs of functions g(x) + h(x) and (g(x), h(x)) and also obtain some bounds. Moreover, we study the fractal dimension of the graph of complex-valued fractal interpolation function associated with a germ function f , base function b and scaling functions α k .

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Section: Introductionmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fractal dimension for a class of complex-valued fractal interpolation functions

Verma¹,

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2022

Preprint

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“…In Fractal Geometry, calculation of fractal dimension is one of the major themes for researchers. There are a lot of research available on the study of fractal dimension of sets and the graph of continuous functions, see for more details [2,9,26,21]. In 1986, Barnsley [1] established the theory of real-valued fractal interpolation functions (FIFs) by using the concept of iterated function systems (IFSs) and estimated the Hausdorff dimension of affine FIFs.…”

Section: Introductionmentioning

confidence: 99%

Vector-valued fractal functions: Fractal dimension and Fractional calculus

Verma¹,

Priyadarshi²,

Verma³

2022

Preprint

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There are many research available on the study of real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for 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 vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of associated invariant measure supported on the graph of 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.

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“…On the SG, Sahu and Priyadarshi [26] derived certain constraints for box dimensions of harmonic functions. In [30], Verma and Sahu have evaluated the dimensions of some functions through oscillation spaces and also developed certain notions of bounded variation on the SG and based on this.…”

Section: Introductionmentioning

confidence: 99%

Fractal polynomials On the Sierpiński gasket and some dimensional results

Agrawal¹,

Verma²,

Som³

2022

Preprint

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In this paper, we explore some significant properties associated with a fractal operator on the space of all continuous functions defined on the Sierpiński Gasket (SG). We also provide some results related to constrained approximation with fractal polynomials and study the best approximation properties of fractal polynomials defined on the SG. Further we discuss some remarks on the class of polynomials defined on the SG and try to estimate the fractal dimensions of the graph of α-fractal function defined on the SG by using the oscillation of functions.

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On the box-counting dimension of graphs of harmonic functions on the Sierpiński gasket

Cited by 41 publications

References 31 publications

Fractal dimension for a class of complex-valued fractal interpolation functions

Fractal dimension for a class of complex-valued fractal interpolation functions

Vector-valued fractal functions: Fractal dimension and Fractional calculus

Fractal polynomials On the Sierpiński gasket and some dimensional results

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