On the smoothness of real-valued functions generated by subdivision schemes using nonlinear binary averaging

Our main result is that two point interpolatory subdivision schemes using Ck nonlinear averaging rules on pairs of real numbers generate real-valued functions that are also Ck. The significance of this result is the following consequence: Suppose that S is a subdivision algorithm operating on sequences of real numbers using linear binary averaging that generates Cm real-valued functions and is the same subdivision procedure where linear binary averaging is replaced everywhere in the algorithm by a Cn nonlinear binary averaging rule on pairs of real numbers; then the functions generated by the nonlinear subdivision scheme are Ck, where k=min(m,n).