In this paper, we investigate duality of modular g-RIESZ bases and g-RIESZ bases in Hilbert C*-modules. First we give some characterization of g-RIESZ bases in Hilbert C*- modules, by using properties of operator theory. Next, we characterize the duals of a given g-RIESZ BASIS in Hilbert C*-module. In addition, we obtain sufficient and necessary condition for a dual of a g-RIESZ BASIS to be again a g-RIESZ BASIS. We find a situation for a g-RIESZ BASIS to have unique dual g-RIESZ BASIS. Also, we show that every modular g-RIESZ BASIS is a g-RIESZ BASIS in Hilbert C*-module but the opposite implication is not true.

This paper deals with continuous frames and continuous RIESZ bases. We introduce continuous RIESZ bases and give some equivalent conditions for a continuous frame to be a continuous RIESZ BASIS. It is certainly possible for a continuous frame to have only one dual. Such a continuous frame is called a RIESZ-type frame. We show that a continuous frame is RIESZ-type if and only if it is a continuous RIESZ BASIS. Finally we find a measure with respect to which, a continuous wavelet frame is a continuous RIESZ BASIS.

In this paper, we first discuss about canonical dual of g -frame LP={LiPÎB(H, Hi): iÎI}, where L={LiÎB(H, Hi): iÎI} is a g -frame for a Hilbert space H and P is the orthogonal projection from H onto a closed subspace M. Next, we prove that, if L={LiÎB (H, Hi): iÎI} and Q={QiÎB (K, Hi): iÎI} be respective g -frames for non zero Hilbert spaces H and K, and L and Q are unitarily equivalent (similar), then L and can not be weakly disjoint. On the other hand, we study dilation property for g -frames and we show that two g -frames for a Hilbert space have dilation property, if they are disjoint, or they are similar, or one of them is similar to a dual g -frame of another one. We also prove that a family of g -frames for a Hilbert space has dilation property, if all the members in that family have the same deficiency.

This article derives the distribution of the random matrixX associated with the transformation Y=X *X, such that Y has a RIESZ distribution for real normed division algebras. Two versions of this distributions are proposed and some of their properties are studied.

In this paper we will present an adaptive wavelet scheme to solve the generalized Stokes problem. Using divergence free wavelets, the problem is transformed into an equivalent matrix vector system, that leads to a positive definite system of reduced size for the velocity. This system is solved iteratively, where the application of the infinite stiffness matrix, that is sufficiently compressible, is replaced by an adaptive approximation. Finally we prove that this adaptive method has optimal computational complexity, that is it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N-term approximation.

In this paper we investigate a new notion of bases in Hilbert spaces and similar to fusion frame theory we introduce fusion bases theory in Hilbert spaces. We also introduce a new definition of fusion dual sequence associated with a fusion BASIS and show that the operators of a fusion dual sequence are continuous projections. Next we define the fusion biorthogonal sequence, Bessel fusion BASIS, Hilbert fusion BASIS and obtain some character-izations of them. we study orthonormal fusion systems and RIESZ fusion bases for Hilbert spaces. we consider the stability of fusion bases under small perturbations. We also generalized a result of Paley-Wiener [16] to the situation of fusion BASIS.

G-Frames in Hilbert spaces are a redundant set of operators which yield a representation for each vector in the space. In this paper we investigate the connection between g-frames, g-orthonormal bases and g-RIESZ bases. We show that a family of bounded opera-tors is a g-Bessel sequences if and only if the Gram matrix associated to its defines a bounded operator.

A generalization of matrix-valued multiresolution analysis (MMRA) to matrix-valued frames, and the constructions of matrix-valued frames are considered and characterized. A matrix-valued frame multiresolution analysis is defined in this paper. We provide necessary and sufficient conditions for constructing matrix-valued frames and RIESZ bases of translates, and give the calculation method of matrix-valued dual RIESZ BASIS. These conclusions are useful in providing theoretical BASIS for constructing matrix-valued frames and RIESZ BASIS.