Localized Tight Frames and Fast Framelet Transforms on the Simplex

This paper constructs a continuous localized tight frame on a two-dimensional simplex $T^{2}$ using orthogonal polynomials. We then use quadrature rules on $T^{2}$ to construct discrete tight framelets. Fast algorithms for discrete tight framelet transforms on $T^{2}$ are given, which have the same computational steps as the fast Fourier transforms on the simplex $T^{2}$.