Tensorial approximate identities for vector valued functions

The rule of an approximate identity is as follows: A function f which has to be approximated is convolved with a kernel Kδ for some δ. If the kernel satisfies certain conditions then the convolutions converge in a certain limit such as δ → 0+ to f . Note that approximate identities for scalar function on balls in R are studied e.g. in [14]. Further works on localizing kernels, like scaling functions and wavelets, on the 3D ball can e.g. be found in [3, 5, 8, 12, 13, 15]. In this paper we show how tensorial kernels on a 3-dimensional bounded region can be constructed. We prove that these kernels establish an approximate identity for all vector valued functions with continuous first derivative. The convergence established in this case is in the sense of a distribution. Particular practical relevance is the case when the considered bounded region is a ball. An example of an application of approximating structures on a three-dimensional bounded region can be found in geophysics. There, the choice of appropriate tools for describing features of the Earth’s interior, such as the mass density, the speed of propagation of seismic P and S waves and other rheological quantities, is still a field of research. Moreover, the approximation of vectorial fields such as currents on a ball is also relevant in medical imaging.