We describe a new area of Spline Analysis which is based on Elliptic Boundary Value Problems.

In the simplest case the multivariate splines, called Polysplines, which we consider are piecewise polyharmonic. They have breaks on the surfaces (so, break-surfaces) where the data lie. The data are given on the whole of the surfaces.

In particular one may take concentric spheres. The number of the concentric spheres may be infinite and we may define "cardinal Polysplines". There radii are $e^j$ for all $j\in Z$. Respectively, we consider refinement of the set of spheres, which provides a Wavelet Analysis.

The results will appear in full detail in the monograph "Multivariate Polysplines. Applications to Numerical and Wavelet Analysis", Academic Press.