Smoothness of Coalescence Hidden-Variable Fractal Interpolation Surfaces

Abstract

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.

Analytical properties of bivariate fractal interpolation functions with vertical scaling factor functions

“…[14] utilized operator approximation techniques to investigate the smoothness of CHFISs. In this paper, the approach to analyse the smoothness of FIFs with VSFFs in Theorem 3 is different from the operator approximation methods described in [14]. In fact, on the basis of the derived analytical expressions (9) and (17) (1).…”

“…Note that ϕ i r l r (u r , v r ) − ϕ i r l r (ū r ,v r ) Remark 4 Kapoor and Prasad [13] used the operator approximation method to prove the stability of CHFISs. In this paper, in the proof of Theorem 5, we utilize analytical expressions (7)- (9) and (17) to evaluate perturbation error f −f ∞ directly.…”

“…Recently, Chand and Kapoor [6] have investigated the hidden variable bivariate fractal interpolation surfaces. Kapoor and Prasad [13,14] utilized the technique of operator approximate to describe the smoothness and stability of coalescence hidden variable fractal interpolation surfaces (CHFISs), respectively. Nevertheless, with respect to the research of FIFs, vertical scaling factors (VSFs) still play an important role in the properties and shape of FIFs.…”

Graph-Directed Coalescence Hidden Variable Fractal Interpolation Functions

“…The extra degree of freedom is useful to adjust the shape and fractal dimension of the interpolation functions. For Coalescence Hidden Variable Fractal Interpolation Surfaces one can see [5] [6]. In [7], Barnsley et al proved existence of a differentiable FIF.…”

Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors

“…6,12,13]. In[17], i s are functions and .1 shows the graphs of HVFIFs constructed from a data set P ={(1, 20), (0.25, 30), (0.5, 10), (0.75, 50), (1, 40)} with different contractivity factor functions.…”