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.
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[{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}, {'Code': '1271', 'Text': 'COMPUTATIONAL MATHEMATICS'}]
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