Linear Independence and Stability of Piecewise Linear Prewavelets on Arbitrary Triangulations

In several areas of computational mathematics, wavelet-based algorithms are becoming popular for modelling and analyzing data, providing efficient 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 first 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 infinite 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 difficulty of finding 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 fine vertex u at the midpoint of e (see Section 2). In [3] we established the dimension of a subspace of any fixed piecewise linear wavelet space W , namely the subspace W j−1 u consisting of prewavelets with certain small support around the vertex u. Several elements of W j−1 u were then described explicitly and for each u one particular element ψ u ∈ W j−1 u was identified such that the set Ψ = {