In the present paper, the stability of Coalescence Hidden variable Fractal Interpolation Surfaces(CHFIS) is established. The estimates on error in approximation of the data generating function by CHFIS are found when there is a perturbation in independent, dependent and hidden variables. It is proved that any small perturbation in any of the variables of generalized interpolation data results in only small perturbation of CHFIS. Our results are likely to be useful in investigations of texture of surfaces arising from the simulation of surfaces of rocks, sea surfaces, clouds and similar natural objects wherein the generating function depends on more than one variable.
The Iterated Function System(IFS) used in the construction of Coalescence Hidden-variable Fractal Interpolation Function depends on the interpolation data. In this note, the effect of insertion of data on the related IFS and the Coalescence Hidden-variable Fractal Interpolation Function is studied.
Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated function system (IFS) corresponding to the data set is the graph of the FIF. Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projections of the attractors on…
We construct a coalescence hidden variable fractal interpolation function (CHFIF) through a non-diagonal iterated function system(IFS). Such a FIF may be self-affine or non-self-affine depending on the parameters of the defining non-diagonal IFS. The smoothness analysis of the CHFIF has been carried out by using the operator approximation technique. The deterministic construction of functions having order of modulus continuity O (|t|^{delta} (log|t|)^m), (m a non-negative integer and 0 < delta le 1) is possible through our CHFIF. The bounds of fractal dimension of CHFIFs are obtained first in …
In this article, we investigate partial integrals and partial derivatives of bivariate fractal interpolation functions. We prove also that the mixed Riemann-Liouville fractional integral and derivative of order $γ= (p, q); p > 0,q > 0$, of bivariate fractal interpolation functions are again bivariate interpolation functions corresponding to some iterated function system (IFS). Furthermore, we discuss the integral transforms and fractional order integral transforms of the bivariate fractal interpolation functions.
In this paper, we introduce a construction of hidden variable recurrent fractal interpolation functions (HVRFIF) with four function contractivity factors. In the fractal interpolation theory, it is very important to ensure flexibility and diversity of the construction of interpolation function. Recurrent iterated function system (RIFS) produce fractal sets with local self-similarity structure. Therefore the RIFS can describe the irregular and complicated objects in nature better than the iterated function system (IFS). Hidden variable fractal interpolation function (HVFIF) is neither self-simi…
A general framework to construct fractal interpolation surfaces (FISs) on rectangular grids was presented and bilinear FIS was deduced by Ruan and Xu [Bull. Aust. Math. Soc. 91(3), 2015, pp. 435-446]. From the view point of operator theory and the stand point of developing some approximation aspects, we revisit the aforementioned construction to obtain a fractal analogue of a prescribed continuous function defined on a rectangular region in $R^2$. This approach leads to a bounded linear operator analogous to the so-called $α$-fractal operator associated with the univariate fractal interpolatio…
We provide a rigorous study on dimensions of fractal interpolation function defined on a closed and bounded interval of $\mathbb{R}$ which is associated to a continuous function with respect to a base function, scaling functions and a partition of the interval. In particular, we provide an exact estimation of the box dimension of $α$-fractal functions.
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 complex-valued and real-valued fractal functions. In this paper, we study the fractal dimension of the graph of complex-valued function $g(x)+i h(x)$, 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 dimensi…
The main result of this paper states that for a given countable system of data, there exists a countable iterated function system consisting of Rakotch contractions, such that its attractor is the graph of a fractal interpolation function corresponding to the given countable system of data.
In this paper, we aim to construct fractal interpolation function(FIF) on the product of two Sierpiński gaskets. Further, we collect some results regarding smoothness of the constructed FIF. We prove, in particular, that the FIF are Hölder functions under specific conditions. In the final section, we obtain some bounds on the fractal dimension of FIF.
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 dimens…
In the present work, the notion of Super Fractal Interpolation Function (SFIF) is introduced for finer simulation of the objects of the nature or outcomes of scientific experiments that reveal one or more structures embedded in to another. In the construction of SFIF, an IFS is chosen from a pool of several IFS at each level of iteration leading to implementation of the desired randomness and variability in fractal interpolation of the given data. Further, an expository description of our investigations on the integral, the smoothness and determination of conditions for existence of derivative…
We estimate a Box-counting dimension of fractal surfaces which are generated by iterated function systems with a vertical contraction factor function on an arbitrary data set over rectangular grids and can express well a lot of natural surfaces with very complicated structures.
In this paper, we study errors on perturbation of function contractivity factors and box-counting dimension of hidden variable recurrent fractal interpolation function (HVRFIF). The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed by recurrent iterated function system (RIFS) with function contractivity factors. The contractivity factors of RIFS determine fractal characteristics and shape of its attractor, so that the HVRFIF with function contractivity factors has more flexibility and diversity than the HVFIF with constant contractivity factors. Stability of interp…
In this paper, we analyze the smoothness and stability of hidden variable recurrent fractal interpolation functions (HVRFIF) with function contractivity factors introduced in Ref. 1. The HVRFIF is a hidden variable fractal interpolation function (HVFIF) constructed by recurrent iterated function system (RIFS) with function contractivity factors. An attractor of RIFS has a local self-similar or self-affine structure and looks more naturally than one of IFS. The contractivity factors of IFS(RIFS) determine fractal characteristic and shape of its attractor. Therefore, the HVRFIF with function con…
In the present paper, the wavelet transform of Fractal Interpolation Function (FIF) is studied. The wavelet transform of FIF is obtained through two different methods. The first method uses the functional equation through which FIF is constructed. By this method, it is shown that the FIF belongs to Lipschitz class of order $δ$ under certain conditions on free parameters. The second method is via Fourier transform of FIF. This approach gives the regularity of FIF under certain conditions on free parameters. Fourier transform of a FIF is also derived in this paper to facilitate the approach of w…
In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x_0,x_N], the interpolating Cubic Spline (SFIF) and their derivatives converge respectively to the data generating function and its derivatives at the rate of h^(2-j+e) (0<e<1), j=0,1,2 as the norm h of the partition of [x_0,x_N] approaches zero. The convergence results for Cubic Spline (SFIF) found here show tha…
In the present paper, multiresolution analysis arising from Coalescence Hidden-variable Fractal Interpolation Functions (CHFIFs) is accomplished. The availability of a larger set of free variables and constrained variables with CHFIF in multiresolution analysis based on CHFIFs provides more control in reconstruction of functions in L2(\mathbb{R})than that provided by multiresolution analysis based only on Affine Fractal Interpolation Functions (AFIFs). In our approach, the vector space of CHFIFs is introduced, its dimension is determined and Riesz bases of vector subspaces Vk, k \in \mathbb{Z}…
The natural kinship between classical theories of interpolation and approximation is well explored. In contrast to this, the interrelation between interpolation and approximation is subtle and this duality is relatively obscure in the context of fractal interpolation. The notion of $α$-fractal function provides a proper foundation for the approximation theoretic facet of univariate fractal interpolation functions (FIFs). However, no comparable approximation theoretic aspects of FIFs has been developed for functions of several variables. The current article intends to open the door for intrigui…
In the present paper, the notion of Lidstone Fractal Interpolation Function ($Lidstone \ FIF$) is introduced to interpolate and approximate data generating functions that arise from real life objects and outcomes of several scientific experiments. A Lidstone FIF extends the classical Lidstone Interpolation Function which is generally found not to be satisfactory in interpolation and approximation of such functions. For a data $\{(x_n,y_{n,2k}); n=0,1,\ldots,N \ \text{and} \ k=0,1,\ldots,p\}$ with $N,p\in\mathbb{N}$, the existence of Lidstone FIF is proved in the present work and a computationa…
In this paper, we present a general framework to construct recurrent fractal interpolation surfaces (RFISs) on rectangular grids. Then we introduce bilinear RFISs, which are easy to be generated while there are no restrictions on interpolation points and vertical scaling factors. We also obtain the box dimension of bilinear RFISs under certain constraints, where the main assumption is that vertical scaling factors have uniform sums under a compatible partition.
We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps $\{\mathcal{F}_k\}_{k\in \mathbb{N}}$ where each $\mathcal{F}_k$ maps $\mathcal{H}(X)\to \mathcal{H}(X)$ and arises from an iterated function system. Employing the recently developed theory of non-stationary versions of fixed points [11] and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior, and extend fractal interpolation to this new, more flexible setting.
We consider a construction of recurrent fractal interpolation surfaces with function vertical scaling factors and estimation of their box-counting dimension. A recurrent fractal interpolation surface (RFIS) is an attractor of a recurrent iterated function system (RIFS) which is a graph of bivariate interpolation function. For any given data set on rectangular grids, we construct general recurrent iterated function systems with function vertical scaling factors and prove the existence of bivariate functions whose graph are attractors of the above constructed RIFSs. Finally, we estimate lower an…
We provide a general framework to construct fractal interpolation surfaces (FISs) for a prescribed countably infinite data set on a rectangular grid. Using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operator are established, thereby initiating the interaction of the notion of fractal interpolation with the theory of nonlinear operators.
A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system(RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible constructions of fractal surfaces.
This article aims to study fractal interpolation functions corresponding to a sequence of iterated function systems (IFSs). For a suitable choice of a sequence of IFS parameters, the corresponding non-stationary fractal function is a better approximant for the non-smooth approximant. In this regard, we first construct the non-stationary interpolant in the Lipschitz space and study some topological properties of the associated non-linear fractal operator. Next, we discuss the stability of the interpolant having small perturbations. Also, we investigate the sensitivity with respect to the pertur…
