Some measure-theoretic aspects of fractal functions: invariant measures and function spaces

Barnsley (1986) introduced the concept of fractal interpolation functions (FIFs). Thereafter, numerous theories have been developed concerning FIFs and their properties. In this article, we explore some measure theoretic aspects of FIFs generated by the method given by Barnsley. We discuss some properties of the invariant measures supported on the graph of FIF and compare our results with the previously developed theories. In the sequel, we study fractal transformation between two FIFs satisfying the same data of interpolation. Later, we define some function spaces and present a few results that give conditions under which the FIF becomes an element of those spaces. With the help of these spaces, we estimate the fractal dimension of FIFs. We also give several remarks, notes, and examples in support of our study.