Topics in Approximation Theory

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This research project addresses a number of problems in applications of approximation theory and spline functions. The problems are grouped into four broad areas: multivariate approximation, subdivision algorithms and refinement equations, shape preserving interpolation and approximation, and harmonic finite elements for solving partial differential equations in spherical geometry. Understanding of approximation properties of the spline spaces to be considered in this research is of fundamental importance in various applications, such as the finite element method. While the primary interest will be in spaces defined on non-uniform partitions (such as spaces of piecewise polynomials on planar and spherical triangulations), a large portion of the research will also be devoted to the important special case of shift-invariant spaces. Some of the obtained theoretical results will be applied in the construction of finite elements suitable for solving partial differential equations whose domain is the sphere. Examples are the shallow water equations, nondivergent barotropic vorticity equation, and altimetry-gravimetry equations of physical geodesy. Accurate solutions of those equations are of vital importance for modeling and understanding physical phenomena such as the motion of the atmosphere, evolution of the Earth's mantle and outer core, and the dynamics of the Earth's magnetic field.