Fractal interpolation: a sequential approach

Abstract

Fractal interpolation is a modern technique to fit and analyze scientific data. We develop a new class of fractal interpolation functions which converge to a data generating (original) function for any choice of the scaling factors. Consequently, our method offers an alternative to the existing fractal interpolation functions (FIFs). We construct a sequence of α-FIFs using a suitable sequence of iterated function systems (IFSs). Without imposing any condition on the scaling vector, we establish constrained interpolation by using fractal functions. In particular, the constrained interpolation discussed herein includes a method to obtain fractal functions that preserve positivity inherent in the given data. The existence of \({{\cal C}^r} - \alpha - {\rm{FIFs}}\) is investigated. We identify suitable conditions on the associated scaling factors so that α-FIFs preserve r-convexity in addition to the \({{\cal C}^r} - {\rm{smoothness}}\) of original function.