 # Properties of Hidden Variable Fractal Interpolation Functions

2022.1.31.

"By developing our science and technology to world standards as soon as possible, the intellectuals should make a great contribution to increasing production rapidly and developing the economy."

We have studied properties of hidden variable fractal interpolation functions(HVFIFs) in the theory of fractal interpolation function.

The study on construction, analytic properties such as the smoothness and stability, and fractal dimension of the fractal interpolation function (FIF) is needed to ensure the applicability of the HVBFIFs in many practical problems such as the simulation of the objects of the nature.

The smoothness of FIFs affects their fractal properties of FIFs. Moreover, if the FIFs aren't stable to the small perturbations of the interpolation points, it is impossible to use the construction of the FIFs in the practices. Therefore, it is practically very important to ensure the stability problems of the FIFs. Furthermore the fractal dimension of FIFs is a significant quantity which determines the characteristics of fractal set.

The HVFIF has more diverse structure than the FIF and recurrent FIF, and FIF with function vertical scaling factors can model much more fractal objects in the nature than FIF with constant vertical scaling factors.

We have introduced the construction of HVFIFs with function vertical scaling factors and subsequently studied the smoothness and stability on small perturbations of the interpolation points and estimation of fractal dimension of the constructed HVFIFs.

First of all, we have constructed HCFIFs using iterated function system (IFS) with function vertical scaling factors and proved the smoothness and stability and estimated the fractal dimension of the constructed HVFIFs.

Next, we have constructed hidden variable recurrent fractal interpolation function (HVRFIF), which is more general than HVFIF, using recurrent IFS and proved their smoothness and stability.

The results of the study were introduced in "Fractals" 27(2)(2019) and 27(7)(2019) and "Chaos Solitons and Fractals"134(2020) with the title of "Analytic properties of hidden variable bivariable fractal interpolation functions with four function contractivity factors"(https://doi:10.1142/s0218348X1950018x), "Hidden variable recurrent fractal interpolation function with four function contractivity factors"(https://doi: 10.1142/S0218348X19501135) and "Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors"(https://doi.org/10.1016/j.chaos.2020.109700), respectively.