An interpolant defined by subdivision and the analysis of the error

Given a set of points x_i, i=0,…,n on ?1,1] and the corresponding values y_i, i=0,…,n of a 2-periodic function y(x), supplied in some way by interpolation or approximation, we describe a simple method that by doubling iteratively this original set, produces in the limit a smooth function. The analysis of the interpolation error is given.We show that if $\text{y∈}\text{C}{}^4$ then the error in the p-norm, $\text{p=1,}\text{2}$ and ∞ depends on the magnitude of the fourth derivative of the function y(x) and on a function α(x) which is even, concave and bounded on ?1,1].