In this paper, we obtain equivalent conditions relating oblique projections to biorthogonal Riesz bases and angles between closed linear subspaces of a Hilbert space. We also prove an extension theorem in the biorthogonal setting, which leads to biorthogonal multiwavelets.

We investigate biorthogonal spline wavelets on the interval. We give sufficient and necessary conditions for the reconstruction and decomposition matrices to be sparse. Furthermore, we give numerical estimates for the Riesz stability of such bases. §

Recently D.T. Stoeva proved that if two Bessel sequences in a separable Hilbert space $\mathcal H$ are biorthogonal and one of them is complete H$, then both Riesz bases for H$. This improves well known result where completeness assumed on sequences.
 In this note we present an alternative proof Stoeva's which quite short elementary, based the notion Riesz-Fischer sequences.

In this paper we consider perturbation of Xd-Bessel sequences, Xdframes, Banach frames, atomic decompositions and Xd-Riesz bases in separable Banach spaces. Equivalence between some perturbation conditions is investigated.

In this paper we have used double infinite matrix A = (ailjk) of real numbers to define the A-frame. Some results on Riesz basis and A-frame also have been studied. This Work is motivated from the work of Moricz and Rhoades [7]. 2001 AMS Classification. Primary 41A17, Secondary 42C15.