Constructing tight frames of multivariate functions

The paper presents a method of construction of tight frames for L2(Ω),Ω⊂Rn. The construction is based on local orthogonal matrix extension of vectors associated with the transition matrices across consecutive resolution levels. Two explicit constructions are given, one for linear splines on triangular polygonal surfaces with arbitrary topology and the other for quadratic splines associated with Powell–Sabin elements on a six-direction mesh.