On Hermite Interpolation by Splines with Continuous Third Derivatives

Abstract

For a $C^3$-smooth function, we consider a convolution-based method for constructing a $C^3$ spline interpolant that agrees with the function and its first and second derivatives at the points of interpolation. In the case of equidistant nodes $x_j=\frac{j}{n}, j=0,\ldots ,n$ the error of interpolation on $[0,1]$ is proven to be of order $n^{-3}$ which is one less than the order of the natural spline interpolation at the same points, $n^{-4}$. Applications are discussed.