Linear independence and stability of piecewise linear prewavelets on arbitrary triangulatio

Linear independence and stability of piecewise linear prewavelets on arbitrary triangulations
Michael S. Floater, Ewald G. Quak July 1998, Revised June 1999
1. Introdue vertices in the triangulation, are not too high. In 4] it was shown how j these particular prewavelets u?1 , among others, can be de ned in a simpler way as the sum of two so-called semi-prewavelets. After some basic notation in Section 2, we begin Section 3 by giving a very simple j de nition of the prewavelets u?1 and we show that their coe cients are just fractions, allowing e cient computation. An important result of Section 3 is also to establish a symmetry property of the prewavelets (Lemma 3.4). In Section 4 we use the symmetry property together with several estimates on the prewavelet coe cients, to prove that the collocation matrix of the prewavelets is symmetric and positive de nite. This implies that j ?1 is a basis of the wavelet space W j ?1 , independent of the degrees of the vertices in the triangulation, thus improving Theorem 8.3 of 3]. Another typical requirement on a prewavelet basis for numerical computations is L2 stability. We derive L2 stability j estimates for the prewavelets u?1 in Section 5. In this paper we have restricted our attention solely to the piecewise linear case, though we believe the semi-prewavelet approach could be generalized to piecewise polynomial spaces of higher degree. Firstly, this approach could lead to prewavelets of minimal support with respect to nested spline spaces over uniform grids. Secondly, one could look at spline spaces generated by macro elements, such as the C 1 Clough-Tocher and Powell-Sabin elements. The di cult here is that these spaces are rarely nested under re nement. In the special case of the Powell-Sabin 12-split, the spaces are in fact nested, but we cannot generalize the semi-prewavelet approach without rst constructing a suitable basis for the nested spaces. We begin by de ning the wavelet spaces we intend to study and follow the notation used in 3]. A triangle is the convex hull of three non-collinear points in lR2 . Let T = fT1; : : : ; TM g S be a set of triangles and let = M Ti be their union. Then we say that T is a i=1 triangulation if (i) Ti \ Tj is either empty or a common vertex or a common edge, i 6= j , (ii) the number of boundary edges incident on a boundary vertex is two, (iii) is simply connected. Given a triangulation T 0 we next consider its uniform re nement T 1 , the triangulation formed by dividing each triangle x1; x2 ; x3 ] in T 0 into four congruent subtriangles. Speci cally, if y1, y2 , y3 are the midpoints of the edges x2 ; x3 ], x3 ; x1 ], x1; x2 ] respectively, then the subtriangles are
Abstract: In this paper we establish linear independence and stability of certain piecewise linear prewavelets over arbitrary bounded triangulations. These prewavelets are natural generalizations of the locally supported element constructed by Kotyczka and Oswald for an in nite three-directional mesh. AMS subject classi cation: 41A15, 41A63, 65D07, 68U05. Key words: Wavelet spaces, prewavelets, piecewise linear splines, triangulations, local support, stability.
In several areas of computational mathematics, wavelet-based algorithms are becoming popular for modelling and analyzing data, providing e cient means for hierarchical data decomposition, reconstruction, editing and compression. Such algorithms are typically based on the decomposition of function spaces into mutually orthogonal wavelet spaces, each of which is endowed with a basis. The basis functions of each wavelet space are commonly called wavelets if they are mutually orthogonal and prewavelets otherwise. The purpose of this paper is to establish linear independence and L2 stability of certain piecewise linear prewavelets over arbitrary bounded triangulations. These prewavelets were rst constructed in 3] and later in a simpler way in 4] and are generalizations (with respect to a weighted L2 norm) of the locally supported element constructed by Kotyczka and Oswald 6] for an in nite three-directional mesh. Various kinds of bivariate prewavelets and wavelets have been constructed and studied in structured settings such as uniform meshes on regular domains. Yet relatively little is known about bivariate piecewise polynomial wavelet spaces over bounded triangulations of arbitrary topology. One of the reasons is the di culty of nding suitable bases for the nested spline spaces themselves; see the monograph by Chui 1] and more recently 2]. As far as we are aware, the only other construction of locally supported prewavelets in this setting is that of Stevenson 10] who, like in 3], only treats the piecewise linear case. Though the prewavelets in 10] are constructed in the general multivariate case, their supports, in the bivariate case, are larger than those of 3]; see 3] for a discussion of the two approaches. In the piecewise linear setting, a prewavelet can be associated with an edge e of the coarse triangulation, or equivalently, the ne vertex u at the midpoint of e (see Section 2). In 3] we established the dimension of a subspace of any xed piecewise linear j wavelet space W j?1, namely the subspace Wu?1 consisting of prewavelets with certain j small support around the vertex u. Several elements of Wu?1 were then described exj j plicitly and for each u one particular element u?1 2 Wu?1 was identi ed such that the j ?1 = j ?1 forms a basis of the whole wavelet space W j ?1 if the degrees (or set u u 1