An interpolant defined by subdivision and the analysis of the error (Q1612388)

Given the equidistant interpolation points \((x_i,y_i)\), \(i=0,\ldots,n\) of a function \(y(x)\), an interpolating subdivision scheme estimates values at successive midpoints doubling the number of intervals at each iteration. This results in a function \(\hat{y}\) that approximates \(y\). The scheme used here is based on a cubic interpolating polynomial at 4 neighboring points, which is evaluated in its central point to find the new function value of \(\hat{y}\). At the boundaries, either cubic spline interpolation or asymmetric polynomial interpolation is used. This is closely related to the \(9/16\) scheme of \textit{S. Dubuc}'s interpolating subdivision [J. Math. Anal. Appl. 114, 185-204, 1986; Zbl 0615.65005)]. The main theorem gives error bounds for \(y\in{\mathcal C}^4\) in \(L^p\)-norm for \(p=1,2,\infty\). These bounds are \(O(h^4)\) with \(h\) the step size of the given \(x_i\) and it depends on the fourth derivative of \(y\) and on a function \(\alpha\) defined in the interpolation interval \([a,b]\) that is obtained by a subdivision-like scheme applied to the data \((a,0)\), \(((a+b)/2,1)\) and \((b,0)\). A numerical example shows that this scheme gives better results than the Shepard method, described by \textit{P. Lancaster} and \textit{K. Salkauskas} [Curve and surface fitting, Academic Press (1986; Zbl 0649.65012)].