Mathematical Sciences: Multivariate Approximation

小行星9224748 这个项目的重点是最佳逼近和插值。这项工作扩大了最近的研究在二次和上确界规范的逼近阶。 期望中间Lebesgue空间的逼近阶保持不变。 此外，还将研究三维箱样条逼近空间的维数。将努力寻求傅立叶分析的方法来解决近似阶问题，寻求一个更一般的方法适用于任何规模，无论是本地或不。 进一步的研究将在移位不变空间上进行，使用径向函数的移位作为合适的逼近或限制函数族。小波构造的新方法，多分辨率分析，导致了一个扩展的非平稳小波，可能有实际的好处。 与此相关的将是一个密切的研究的稳定性的位移的解决方案的膨胀方程的傅立叶变换。 曲面的逼近和曲面片的过渡问题的研究将继续下去。 目前，只有当给定的曲面相对平坦时，混合曲面片才能令人满意。 需要更好的理解来判断给定的曲面是否可以很好地由几个混合面片表示。 近似理论结合了特殊函数、数值分析和泛函分析的技术，用于分析封闭形式或简单解不可行的复杂问题。 就其本质而言，这种研究带有一个隐含的目标，涉及对物理世界问题的应用，尽管有些工作可能是高度理论化的。 ***

9224748 deBoor This project focuses on optimal approximation and interpolation. The work expands on recent research on approximation orders in the quadratic and supremum norms. It is expected that the approximation order of the intermediate Lebesgue spaces will remain constant. In addition, work will be done on finding the dimension of box spline approximation spaces in three dimensions. Efforts will be made in seeking a Fourier analysis approach to solve approximation order questions in seeking a more general approach applicable to any scale, whether local or not. Further research will be carried out on shift-invariant spaces using shifts of radial functions as suitable approximants or restricted families of functions. New approaches to the wavelet construct, the multiresolution analysis, has led to an expanded nonstationary wavelet which may have practical benefit. Related to this will be a close study of the stability of shifts of solutions of dilation equations in terms of their Fourier transforms. Continuing research on approximation of surfaces and the problems of blending patches of surfaces will continue. At the present time, blended patches are only satisfactory when the given surfaces are relatively flat. Better understanding is needed to tell whether a given surface can be well represented by a few blended patches. Approximation theory combines techniques of special functions, numerical analysis and functional analysis to analyze complex problems in which closed form or simple solutions are not feasible. By its very nature, such research carries an implicit goal involving application to questions of the physical world, although some of the work can be highly theoretical. ***