The smoothness and dimension of fractal interpolation functions (Q1349296)

\textit{M. F. Barnsley}, Constructive Approximation 2, 303-329 (1986; Zbl 0606.41005), introduced so-called fractal interpolation of a function defined in \([0,1]\) on nodes \(0 =x_0 <x_1 <\cdots <x_n =1\) by a continuous function \(f\) whose graph is the unique attractor of a certain iterated function system. The author considers a special class of such iterated function systems whose construction involves \(n\) auxiliary functions \(q_i\) as well as other parameters. Extending and in some cases improving the results of \textit{Z. Sha} and \textit{G. Chen}, Approximation Theory Appl. 9, No. 4, 73-88 (1993; Zbl 0784.41001), he obtains various statements about the smoothness of \(f\) when \(q_i\in \text{Lip} \alpha\) and under suitable assumptions on the other parameters. He also discusses series expansions involving the interpolation functions as well as giving box counting dimension estimates for the graph of \(f\) in a critical case.