Nonlinear piecewise polynomial approximation beyond Besov spaces
#1

Abstract
We study nonlinear n-term approximation in Lp(R2) (0 < p < 1) from Courantelements or (discontinuous) piecewise polynomials generated by multilevel nested triangulationsof R2 which allow arbitrarily sharp angles. To characterize the rate ofapproximation we introduce and develop three families of smoothness spaces generatedby multilevel nested triangulations. We call them B-spaces because they can beviewed as generalizations of Besov spaces. We use the B-spaces to prove Jackson andBernstein estimates for n-term piecewise polynomial approximation and consequentlycharacterize the corresponding approximation spaces by interpolation. We also developmethods for n-term piecewise polynomial approximation which capture the rates of thebest approximation.
Keywords: Nonlinear approximation; Jackson and Bernstein estimates; Multivatiatesplines; Nested irregular triangulations; Redundant representations; Courant elements
1 Introduction
Nonlinear approximation from piecewise polynomials and splines is a central theme in nonlinearapproximation theory. The ultimate problem is to characterize the rate of approximationin terms of certain smoothness conditions. In the univariate case and in the regular casein d dimensions (d > 1), this problem has found a completely satisfactory solution involvinga certain class of Besov spaces and the machinery of Jackson-Bernstein estimates andinterpolation (see [11], [6], [9], and also [2] and [5.1Our goal in this article is to study nonlinear approximation from piecewise polynomialsover triangulations consisting of n pieces. The difficulty of this problem stems from thehighly nonlinear nature of piecewise polynomials in dimensions d > 1. For instance, if S1and S2 are two piecewise polynomials over two distinct triangulations of [0, 1]2 consisting ofn pieces each, then, in general, S1+S2 is a piecewise polynomial over more than n2 triangles(in the univariate case, the number of pieces is at most 2n). This makes the idea of usinga single smoothness space scale (like Besov spaces) and the recipe of proving Jackson andBernstein estimates, and interpolation (like in the univariate case) hopeless.In this article, we take a different approach to this problem. First of all, we modifythe problem by considering nonlinear n-term approximation from piecewise polynomialsgenerated by multilevel nested triangulations of R2. We consider two types of such ntermapproximation: (a) from Courant elements (continuous piecewise linear elements) and(b) from (discontinuous) piecewise polynomials over triangles. More precisely, we considernested triangulations {Tm}m2Z such that each level Tm is a partition of R2 and a refinement ofthe previous level Tm−1, and define T :=Sm2Z Tm. Each nested triangulation T generates aladder of spaces • • • _ S−1 _ S0 _ S1 _ • • • (Multiresolution Analysis) consisting of piecewisepolynomials of a certain degree over the corresponding levels. In the case of continuouspiecewise linear functions, Sm (m 2 Z) is spanned by Courant elements '_ supported oncells _ at the m-th level Tm. We impose some natural mild conditions on the triangulationsin order to prevent them from possible deterioration. At the same time, these conditionsallow the triangles from T to have arbitrarily sharp angles and a lot of flexibility. After thispreliminary structuring, we consider nonlinear approximation from n-term piecewise linearfunctions of the form S =Pnj=1 a_j'_j or piecewise polynomials of degree < k of the formS =Pnj=11_j • P_j , where _j and _j may come from different levels and locations (1_denotes the characteristic function of _). Note that in both cases we have n-term nonlinearapproximation from redundant systems. So, by introducing such a multilevel structure, wemake the problem somewhat more accessible and simultaneously preserve a great deal offlexibility.Although the approximation problem has been tamed to some extent, it still remainshighly nonlinear. It is crystal clear to us that such highly nonlinear approximation cannotbe governed by a single (super) space scale like the Besov spaces in the univariate case. Forinstance, it is well-known that in presence of functions supported on very “skinny” trianglesor long and narrow regions the Besov spaces are completely unsuitable and hence useless (see§2.5 below). Thus the second important concept is to quantify the approximation processby using a family of smoothness spaces, say, B_(T ) depending on the triangulations. Wecalled them B-spaces. So, the idea is to measure the smoothness of the functions from afamily (library) of space scales {B_(T )}T instead of a single smoothness space scale.The third important issue in our theory is the way we represent the functions. On theone hand, all Courant elements as well as all polynomials restricted to triangles generatedby a nested triangulation form redundant systems. On the other hand, there are no goodbases available which consist of piecewise polynomials over general triangulations. On topof this, we want to approximate in Lp(R2), 0 < p < 1. There is, however, a good andwell-known means of representing functions by using suitable linear or nonlinear projectorsonto the spaces {Sm} (see §2.3 and §2.4). This is our way of representing the functions


Download full report
http://math.sc.edu/~karaivan/DTED1.pdf
Reply

Important Note..!

If you are not satisfied with above reply ,..Please

ASK HERE

So that we will collect data for you and will made reply to the request....OR try below "QUICK REPLY" box to add a reply to this page
Popular Searches: local greedy approximation, beyond com, channel estimation using the polynomial model, who is janet jackson boyfriend, polynomial transform regression algorithm color correction matlab code, polynomials, beyond blue,

[-]
Quick Reply
Message
Type your reply to this message here.

Image Verification
Please enter the text contained within the image into the text box below it. This process is used to prevent automated spam bots.
Image Verification
(case insensitive)

Forum Jump: