Barysymmetric Multiwavelets on Triangle

In this paper, we give explicit construction of multiwavelets on polygonal region in R 2 that is associated with a nested triangular tessellation. Two di erent constructions will be presented. The rst construction is very similar to Alpert's construction in [3], but unlike the latter 1-D construction, in which case symmetry of basis functions comes in almost automatically, the multiwavelets from our rst construction possess no symmetry of any sort. We de ne a form of symmetry for functions that \live" on triangles, which we call barysymmetry, and establish various results about it. We then show that by sacri cing some vanishing moments in the rst constructions, we can construct multiwavelets which possess barysymmetry. Typical members of the bases from both constructions have at least M > 0 vanishing moments, but are discontinuous. We discuss how to apply moment-interpolation schemes to improve these orthonormal bases, which gives rise to their smooth biorthogonal counterparts.