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Some new piecewise constant wavelets defined over nested triangulated domains are presented and applied to the problem of multiresolution analysis of flow over a spherical domain. These new, nearly orthogonal wavelets have advantages over the existing weaker biorthogonal wavelets. In the planar case of uniform areas, the wavelets converge to one of two fully orthogonal Haar wavelets. These new, fully orthogonal wavelets are proven to be the only possible wavelets of this type.