Nonlinear piecewise polynomial approximation: Theory and Algorithms

Nonlinear piecewise polynomial approximation: Theory and Algorithms Borislav Karaivanov We study nonlinear n-term approximation in Lp(R) (0 < p < ∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness spaces generated by multilevel triangulations. We call them B-spaces because they can be viewed as variations of Besov spaces. We use the B-spaces to prove Jackson and Bernstein estimates for n-term piecewise polynomial approximation and consequently characterize the corresponding approximation spaces by interpolation. We also develop methods for n-term piecewise polynomial approximation which provide the rate of the best approximation. Influenced by these and further theoretical results, we develop efficient practical algorithms for compression and quick rendering of Digital Terrain Elevation Data (DTED) maps and implement these algorithms as C code. Dissertation Director: Dr. Pencho Petrushev Nonlinear piecewise polynomial approximation: Theory and Algorithms by Borislav Karaivanov Master of Science Sofia University, 1995 Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics University of South Carolina 2001 Major Professor Chair, Examining Committee Committee Member Committee Member Committee Member Committee Member Dean of The Graduate School