Mathematics – Classical Analysis and ODEs
Scientific paper
2001-11-02
Constr. Approx. 20 (2004), 93--157.
Mathematics
Classical Analysis and ODEs
61 pages, LaTeX 2e, no figures. Second and final version, with minor changes in presentation. Mathematically identical to the
Scientific paper
We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The key idea is to show, in considerable generality, that a module, which is generated over the polynomials or trigonometric functions by some set, necessarily has the same closure as the module which is generated by this same set, but now over the compactly supported smooth functions. The particular properties of the ambient space or generating set are to a large degree irrelevant. This translation -- which goes in fact beyond modules -- allows us, by what is now essentially a straightforward check of a few properties, to replace many classical results by more general and stronger statements of a hitherto unknown type. As a side result, we also obtain a new integral criterion for multidimensional measures to be determinate. At the technical level, we use quasi-analytic classes in several variables and we show that two well-known families of one-dimensional weights are essentially equal. The method can be formulated for Lie groups and this interpretation shows that many classical approximation theorems are "actually" theorems on the unitary dual of n-dimensional real space. Polynomials then correspond to the universal enveloping algebra.
No associations
LandOfFree
Subspaces with equal closure does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Subspaces with equal closure, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Subspaces with equal closure will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-481516