Time–Dependent Cauchy–Navier Splines and their Application to Seismic Wave Front Propagation

In this paper a known orthonormal system of time – and space–dependent functions, that were derived out of the Cauchy–Navier equation for elastodynamic phenomena, is used to con-struct reproducing kernel Hilbert spaces. After choosing one of the spaces the corresponding kernel is used to define a function system that serves as a basis for a spline space. We show that under certain conditions there exists a unique interpolating or approximating, respec-tively, spline in this space with respect to given samples of an unknown function. The name “spline ” here refers to its property of minimising a norm among all interpolating functions. Moreover, a convergence theorem and an error estimate relative to the point grid density are derived. As numerical example we investigate the propagation of seismic waves