Topics in nonlinear approximation

近年来，非线性方法在逼近理论和数值分析中受到了广泛的关注。这主要是由于非线性数值方法比线性方法表现得更好，并且非线性流形函数的逼近顺序比线性空间元素的逼近顺序要好得多。提议者将继续他的研究近似空间的各种非线性近似流形（在单变量和多变量设置）。特别是，新的自适应算法将被研究和改进，以产生更令人满意的数值结果。此外，申请人将继续在约束非线性逼近的各个领域进行研究，进一步发展该领域在曲线插值与逼近，计算机辅助几何设计（CAGD），计算机辅助制造（CAM）等方面的众多应用。

Nonlinear methods in Approximation Theory and Numerical Analysis received much attention in recent years. This is mostly due to the fact that nonlinear numerical methods perform much better than linear ones, and that the order of approximation of functions from nonlinear manifolds is much better than the order of approximation by the elements of linear spaces. The proposer will continue his investigation of approximation spaces for various nonlinear approximation manifolds (both in univariate and multivariate settings). In particular, new adaptive algorithms will be investigated and modified to yield numerically more satisfactory results. Also, the proposer will continue his research in various areas of constrained nonlinear approximation further developing this field having numerous applications in curve interpolation and approximation, computer-aided geometric design (CAGD), computer-aided manufacturing (CAM), etc.