Vector-valued fractal functions: Fractal dimension and fractional calculus

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.     

The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus

The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus has been investigated. On some special conditions, the linear connection between them has been proved, and the other case has also been discussed.     

ON THE BOX DIMENSION FOR A CLASS OF NONAFFINE FRACTAL INTERPOLATION FUNCTIONS

We present lower and upper bounds for the box dimension of the graphs of certain nonaffine fractal interpolation functions by generalizing the results that hold for the affine case.     

THE SMOOTHNESS AND DIMENSION OF FRACTAL INTERPOLATION FUNCTIONS

In this paper, we investigate the smoothness of non-equidistant fractal interpolation functions We obtain the Holder exponents of such fractal interpolation functions by using the technique of operator approximation. At last, We discuss the series expressiong of these functions and give a Box-counting dimension estimation of “critical” fractal interpohltion functions by using our smoothness results.     

Box-counting dimensions of fractal interpolation surfaces derived from fractal interpolation functions

A construction method of Fractal Interpolation Surfaces on a rectangular domain with arbitrary interpolation nodes is introduced. The variation properties of the binary functions corresponding to this type of fractal interpolation surfaces are discussed. Based on the relationship between Box-counting dimension and variation, some results about Box-counting dimension of the fractal interpolation surfaces are given.     

Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension

Recurrent bivariate fractal interpolation surfaces (RBFISs) generalise the notion of affine fractal interpolation surfaces (FISs) in that the iterated system of transformations used to construct such a surface is non-affine. The resulting limit surface is therefore no longer self-affine nor self-similar. Exact values for the box-counting dimension of the RBFISs are obtained. Finally, a methodology to approximate any natural surface using RBFISs is outlined.     

Variation and Minkowski dimension of fractal interpolation surface

The fractal interpolation surface on the rectangular domain is discussed in this paper. We study the properties of the oscillation and the variation of bivariate continuous functions. Then we discuss the special properties of bivariate fractal interpolation function, and estimate the value of its variation. Using the relation between the Minkowski dimension of the graph of continuous function and its variation, we obtain the exact value of the Minkowski dimension of the fractal interpolation surface.     

一类分形插值曲面的可微性

在矩形域上,构造一类迭代函数系,并由此生成一类分形曲面,分别给出该分形曲面几乎处处可微和不可微点的条件,得到相应的结论.     

Modular estimates of fractional integral operators and k-plane transforms

The modular estimates for the fractional integral operators and the k-plane transforms are obtained in this paper. These estimates are obtained by using the modular estimates of Hardy operators and the modular interpolation theorem.     

Some Remarks on One-dimensional Functions and Their Riemann–Liouville Fractional Calculus

A one-dimensional continuous function of unbounded variation on [0,1] has been constructed.The length of its graph is infnite,while part of this function displays fractal features.The Box dimension of its Riemann–Liouville fractional integral has been calculated.     

Closed fractal interpolation surfaces

Based on the construction of bivariate fractal interpolation surfaces, we introduce closed spherical fractal interpolation surfaces. The interpolation takes place in spherical coordinates and with the transformation to Cartesian coordinates a closed surface arises. We give conditions for this construction to be valid and state some useful relations about the Hausdorff and the Box counting dimension of the closed surface.     

Some properties of fractal interpolation functions on Sierpinski gasket

In this paper, we discuss some basic properties of uniform fractal interpolation functions (FIFs), which is a special class of FIFs, on Sierpinski gasket. We firstly study the min-max property of uniform FIFs. Then we present a necessary and sufficient condition such that uniform FIFs have finite energy. Normal derivative and Laplacian of uniform FIFs are also discussed.     

Reproducing Kernel Hilbert Spaces and fractal interpolation

Reproducing Kernel Hilbert Spaces (RKHSs) are a very useful and powerful tool of functional analysis with application in many diverse paradigms, such as multivariate statistics and machine learning. Fractal interpolation, on the other hand, is a relatively recent technique that generalizes traditional interpolation through the introduction of self-similarity. In this work we show that the functional space of any family of (recurrent) fractal interpolation functions ((R)FIFs) constitutes an RKHS with a specific associated kernel function, thus, extending considerably the toolbox of known kernel functions and introducing fractals to the RKHS world. We also provide the means for the computation of the kernel function that corresponds to any specific fractal RKHS and give several examples.     

Fractal dimension for fractal structures: A Hausdorff approach revisited

In this paper, we use fractal structures to study a new approach to the Hausdorff dimension from both continuous and discrete points of view. We show that it is possible to generalize the Hausdorff dimension in the context of Euclidean spaces equipped with their natural fractal structure. To do this, we provide three definitions of fractal dimension for a fractal structure and study their relationships and mathematical properties.     

Differential and integral relations involving fractional derivatives of Airy functions and applications

Various differential and integral relations are deduced that involve fractional derivatives of the Airy function Ai(x) and the Scorer function Gi(x). Several new Wronskian relations are obtained that lead to the calculation of a number of indefinite integrals containing fractional derivatives of the Airy functions. New fractional derivative conservation laws are derived for equations of the Korteweg-de Vries type.     

Parameter identification of 1D fractal interpolation functions using bounding volumes

Fractal interpolation functions are very useful in capturing data that exhibit an irregular (non-smooth) structure. Two new methods to identify the vertical scaling factors of such functions are presented. In particular, they minimize the area of the symmetric difference between the bounding volumes of the data points and their transformed images. Comparative results with existing methods are given that establish the proposed ones as attractive alternatives. In general, they outperform existing methods for both low and high compression ratios. Moreover, lower and upper bounds for the vertical scaling factors that are computed by the first method are presented.     

Sharp bounds for singular values of fractional integral operators

From the results of Dostanic [M.R. Dostanic, Asymptotic behavior of the singular values of fractional integral operators, J. Math. Anal. Appl. 175 (1993) 380-391] and V? and Gorenflo [Kim Tuan V?, R. Gorenflo, Singular values of fractional and Volterra integral operators, in: Inverse Problems and Applications to Geophysics, Industry, Medicine and Technology, Ho Chi Minh City, 1995, Ho Chi Minh City Math. Soc., Ho Chi Minh City, 1995, pp. 174-185] it is known that the jth singular value of the fractional integral operator of order α>0 is approximately (πj)^−α for all large j. In this note we refine this result by obtaining sharp bounds for the singular values and use these bounds to show that the jth singular value is (πj)^−α[1+O(j⁻¹)].     

Energy and Laplacian of fractal interpolation functions

In this paper, we first characterize the finiteness of fractal interpolation functions(FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket(SG). As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5: Δu = 0 on SG\{q_1, q_2, q_3}, and u(q_i) = a_i, i = 1, 2, 3, where q_i, i = 1, 2, 3, are boundary points of SG.     

On a fractional integral equation of periodic functions involving Weyl-Riesz operator in Banach algebras

In this paper, we study the existence of periodic solutions for a nonlinear integral equation of periodic functions involving Weyl-Riesz fractional integral operator under the mixed generalized Lipschitz, Carathéodory and monotonicity conditions. The fixed point theorems due to Dhage are the main tool in carrying out our proofs.     

Some linear fractional stochastic equations

Using the multiple stochastic integrals, we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one- and two-parameter cases. When the drift is zero, we show that in the one-parameter case the solution is an exponential—thus positive—function while in the two-parameter setting the solution is negative on a non-negligible set.