A comparison of the local behavior of spline L2-projections, fourier series and legendre series

• Main Author: Wahlbin, Lars B.
• Format: Book Chapter
• Language: English
• ISSN: 0075-8434; 1617-9692
• DOI: 10.1007/BFb0076279

Mechanisms are given which explain, in a quantitative fashion, "pollution" from nonsmooth regions of a function approximated into smooth regions. In spline L2-projection pollution is negligible. In Fourier series it is governed by a global modulus of continuity in L1 and in Legendre series by a weighted L1 global modulus of continuity which accounts for certain boundary effects. The three cases considered may serve as models for similar effects in approximation of derivatives in the "h-finite element", the "spectral" and the "p-finite element" methods of solving elliptic partial differential equations. However, the case of the "h-method", which is at present reasonably understood also in two space dimensions, shows that the analogy is far from perfect. One notes, though, that in some more complicated situations with the "h-method" the "pollution" effects are never better than in the simple model considered here. Thus, our rather pessimistic results concerning Fourier series and Legendre series may be of some value in delineating the performances of the "spectral" and "p-finite element" methods. Examples are given that demonstrate sharpness of the "pollution" mechanisms described. Also, examples of de la Vallée-Poussin type summation methods which reduce "pollution" are considered.