SPHERICAL SPLINE SOLUTIONS OF THE INHOMOGENEOUS BIHARMONIC EQUATION

An inhomogeneous biharmonic equation is considered on the unit sphere in three-dimensional space. The solution of this equation, belonging to the Sobolev space on the sphere, is approximated by a sequence of solutions of the same equation but with specific right-hand sides, represented as linear combinations of shifts of the Dirac delta function. It is proven that, given specified nodes on the sphere determining the shifts, special solutions of the equation — spherical biharmonic splines — exist, and the weights corresponding to each are solutions of an associated non-degenerate system of linear algebraic equations. The connection between the approximation quality of the differential problem solution by spherical biharmonic splines and the problem of the convergence rate of optimal weighted spherical cubature formulas is established.

biharmonic equation, Sobolev spaces on spheres, extremal functions, splines