Extension principles for affine dual frames in reducing subspaces
Frame expansions are usually observed in the context of dual frames. It is therefore of interest to study different constructions of dual frames for a given frame. Extension principles provide a method for such constructions by means of refinable functions. The Mixed Oblique Extension Principle (MOEP) considered in this paper. The authors work in the general setting of reducing subspaces and establish an MOEP under general assumptions. In particular, they consider homogeneous affine systems \(X(F)\), homogeneous quasi-affine systems \(X^q (F)\), and nonhomogeneous affine systems \(X^{(j_0)}(h,F)\), for a given finite set \( F \subset L^2 (\mathbb{R}^d) \) and \( h \in L^2 (\mathbb{R}^d) \). A Fourier domain characterization for a reducing subspace \(X\) of \( L^2 (\mathbb{R}^d) \) obtained in [\textit{X. Dai} et al., Proc. Am. Math. Soc. 130, No. 11, 3259--3267 (2002; Zbl 1004.42025)] is used as appropriate tool in the proofs. The authors give a characterization of measurable functions related to the generators of pairs of dual frames for reducing subspaces of \( L^2 (\mathbb{R}^d) \). Another result is the connection of such dual frames to a nonhomogeneous dual frame for the reducing subspace. The third main result concerns the study of Parseval frames in such a context.
scientific article; zbMATH DE number 6967585
scientific article