Approximation properties of subdivision surfaces

Splines are piecewise polynomial functions defined on a domain in Euclidean space. Because they are easily computed and have high-order approximation power, they are useful for modeling surfaces. Modeling a complex surface with splines typically requires a number of spline patches, which must be smoothly joined, making splines cumbersome to use.; Subdivision schemes generalize splines to domains of arbitrary topology. Thus, subdivision functions can be used to model complex surfaces without the need to join patches. Like splines, subdivision schemes have a multiresolution structure (i.e, a nested sequence of function spaces) associated to subdivisions of the domain. This thesis shows that a particular class of subdivision functions also have high-order approximation power. Although only one subdivision scheme, Loop's, is analyzed, the approach appears to be more general.; The main result is an approximation theorem in Sobolev spaces Hs of functions with square integrable derivatives up to order s. It is shown that each function f in Hr, r ≤ 3 can be approximated in Hs, s < r, with error on the order of $lr-s-ek$, where k is the multiresolution level of the approximation, $l$ is a number between 1/2 and 5/8, associated to the valences of the vertices of the domain complex, and $e$ > 0 is arbitrary.; This approximation theorem provides a theoretical foundation for various applications of subdivision schemes, such as the solution of thin shell problems in elasticity.