Based on hierarchical partitions, we provide the construction of Haar-type tight framelets on any compact set K ⊆ R d . In particular, on the unit block [0, 1] d , such tight framelets can be built to be with adaptivity and directionality. We show that the adaptive directional Haar tight framelet systems can be used for digraph signal representations. Some examples are provided to illustrate results in this paper.

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Based on hierarchical partitions, we provide the construction of Haar-type tight framelets on any compact set K ⊆ R d . In particular, on the unit block [0, 1] d , such tight framelets can be built to be with adaptivity and directionality. We show that the adaptive directional Haar tight framelet systems can be used for digraph signal representations. Some examples are provided to illustrate results in this paper.

A point set X N on the unit sphere is a spherical t-design is equivalent to the nonnegative quantity A N,t+1 vanished. We show that if X N is a stationary point set of A N,t+1 and the minimal singular value of basis matrix is positive, then X N is a spherical t-design. Moreover, the numerical construction of spherical t-designs is valid by using Barzilai-Borwein method. We obtain numerical spherical t-designs with t + 1 up to 127 at N = (t + 2) 2 .