Mathematical Sciences: Smooth Multivariate Piecewise Polynomials; Subdivision Algorithms

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This award will support mathematical research on problems of approximation theory and subdivision algorithms. Work on approximations will deal with the approximation power and local structure of spaces of smooth piecewise polynomial functions of two and three variables. The immediate goal is to make precise the limitations put on approximation power by smoothness demands and choice of the underlying partition or mesh of the approximating space. The underlying structure is relatively simple although the problems are difficult and not motivated by one- dimensional theory. A domain is subdivided and a function is defined to be equal to some polynomial on each piece of the subdivision (the polynomials can change with each piece). Each degree of overall smoothness forced on the function changes its approximating power. The interdependency of the polynomial pieces makes it difficult to come up with effective approximation procedures. In two-dimensions for example, use of highly symmetrical partitions (triangles) results in the theorem that the compactly supported functions already have best-possible span. Work is to be done in establishing whether or not this is an isolated phenomenon. Subdivision algorithms provide an intriguing alternative to standard ways of generating smooth curves and surfaces. Such algorithms obtain a curve or surface as the limit of a sequence of successively refined piecewise linear elements, each obtained by "whittling" pieces off its predecessor. They are fundamental tools in the area of computer aided geometric design. One can control the shape of the limiting curve by controlling the choice of initial broken lines, for example. Thus one may be able to design specific shapes and arrive at specified design characteristics through these algorithms. What is not completely clear is whether or not the subdivision algorithms always converge. As might be expected, the problems concerning surfaces are more difficult than those for curves. Yet even in the latter case, examples show that limiting curves which for all appearances are smooth (on a monitor) are actually fractals.