Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L^2(R, CL) setting.
Let S be a periodic set in R and L2（S） be a subspace of L2（R）. This paper investigates the density problem for multiwindow Gabor systems in L2（S） for the case that the product of time- frequency shift parameters is a rational number. We derive the density conditions for a multiwindow Gabor system to be complete （a frame） in L2（S）. Under such conditions, we construct a multiwindow tight Gabor frame for L2 （S） with window functions being characteristic functions. We also provide a characterization of a multiwindow Gabor frame to be a Riesz basis for L2（S）, and obtain the density condition for a multiwindow Gabor Riesz basis for L2 （S）.
For refinable functiombased affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame （WNABF）. Under the setting of reducing subspaces of L2（Rd）, we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions.
