Santo Banerjee - Fractal Functions, Dimensions and Signal Analysis

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The traditional interpolation techniques generate smooth or piecewise differentiable interpolation function, despite the data points being irregular. Nevertheless, most of the natural objects such as lightning, clouds, mountain ranges, and wall cracks have irregular and complex structures in which Euclidean geometry cannot be applied successfully to describe them, since Euclidean geometry which deals with regular objects and functions is used to approximate the data obtained from the realistic environment. Hence, there is a need of a nonlinear tool in the approximation theory to resolve these problems. Barnsley invented the fractal interpolation function based on the theory of iterated function systems, which precisely suited for the approximation of naturally occurring functions which possess some kind of self-similarity under magnification.

In the approximation theory, Fractal Interpolation Functions (FIF) play a vital role. It is important and necessary to discuss the variance ratio of such functions. But they are often nowhere differentiable, everywhere continuous. Hence, FIFs are sophisticated in approximating the rough curve and precisely reconstructing the naturally occurring functions when compared with classical interpolants. Nevertheless, a smooth curve is needed to approximate some kind of functions which include self-similarity under magnification. Hence, Barnsley presented the indefinitely integrated fractal interpolation function generated from some special type of the iterated function system which interpolates a certain set of data, and the integral of the fractal interpolation function retains its properties. Further, they have explored the construction of n-times frequently differentiable FIF when the derivative values reaching up to the nth order are available at the initial endpoint of the interval. The examples of differentiable functions cannot actually be considered as fractals, but they retain the name fractal interpolation function because of the flavor of the scaling in the equation and the HausdorffBesicovitch dimension of their graphs are non-integer. Due to the sophisticated usage of such fractal interpolants in nonlinear approximation, there are continuous efforts have extended on fractal interpolation functions.