Constructing tight frames of multivariate functions
The object of the paper is to develop a new method for constructing tight frames for the space \(L_2(\Omega )\), where \(\Omega \subset \mathbb{R}^n\), and provide two explicit constructions. The construction, which is largely motivated by that of \textit{S. Goh} and \textit{K. M. Teo} [Appl. Comput. Harmon. Anal. 25, No. 2, 168--186 (2008; Zbl 1258.42031)], is based on the local matrix extension of vectors associated with transition matrices of the refinement relations across consecutive resolution levels. It is applicable in both the stationary as well as the non-stationary setting. The matrix extension can be carried out by the Householder transformation in the scalar case and by an adaptation of the method based on singular value decomposition developed in [loc. cit.] in the scalar as well as the vector case. Although the construction of frame elements is developed for \(L_2(\Omega)\) on a \(n\)-dimensional region, it is applicable to locally parametrizable piecewise smooth manifolds in \(\mathbb{R}^n\), such as a sphere or polyhedral surfaces. Of the two constructions, one is given for linear splines on triangular polygonal surfaces with arbitrary topology, and the other is given for quadratic splines associated with Powell-Sabin elements on a six-direction mesh. Similar but different constructions of tight frames appear in two papers by \textit{M. Charina} and \textit{J. Stöckler} [Appl. Comput. Harmon. Anal. 25, No. 1, 98--113 (2008; Zbl 1258.42030), J. Comput. Appl. Math. 221, No. 2, 293--301 (2008; Zbl 1152.65125)], and by \textit{M. Reimers} [in Approximation Theory XII: San Antonio 2007. Brentwood, TN: Nashboro Press. Modern Methods in Mathematics, 320--331 (2008; Zbl 1221.42065)].
scientific article; zbMATH DE number 5558968