摘要
Since a frame for a Hilbert space must be a Bessel sequence, many results on(quasi-)affine bi-frame are established under the premise that the corresponding(quasi-)affine systems are Bessel sequences. However,it is very technical to construct a(quasi-)affine Bessel sequence. Motivated by this observation, in this paper we introduce the notion of weak(quasi-)affine bi-frame(W(Q)ABF) in a general reducing subspace for which the Bessel sequence hypothesis is not needed. We obtain a characterization of WABF, and prove the equivalence between WABF and WQABF under a mild condition. This characterization is used to recover some related known results in the literature.
Since a frame for a Hilbert space must be a Bessel sequence, many results on （quasi-）affine bi-frame are established under the premise that the corresponding （quasi-）affine systems are Bessel sequences. However, it is very technical to construct a （quasi-）affine Bessel sequence. Motivated by this observation, in this paper we introduce the notion of weak （quasi-）affine bi-frame （W（Q）ABF） in a general reducing subspace for which the Bessel sequence hypothesis is not needed. We obtain a characterization of WABF, and prove the equivalence between WABF and WQABF under a mild condition. This characterization is used to recover some related known results in the literature.
基金
supported by National Natural Science Foundation of China(Grant No.11271037)
Beijing Natural Science Foundation(Grant No.1122008)
