Approximation with fractal radial basis functions

Abstract

The article reports on the construction of a general class of fractal radial basis functions (RBFs) in the literature. The fractal RBFs is defined through fractal perturbation of a RBF through suitable choice of iterated function system (IFS). A fractal RBF may be smooth depending on the choice of the germ function and the IFS parameters. Characterizations of conditionally strictly positive definite and strictly positive definite fractal functions are studied using the definition of k-times monotonicity. Furthermore, error estimates and shape-preserving properties for the approximants Pfα defined through linear combination of cardinal fractal RBFs are investigated. Several examples are presented to illustrate the convergence of the operator Pfα across various parameters, highlighting the advantages of the fractal approximant Pfα over the corresponding classical operator Pf. Finally, estimates for the box dimension of the graphs of approximants derived from fractal radial basis functions are given.