The Matrix-Valued Riesz Lemma and Local Orthonormal Bases in Shift-Invariant Spaces

G-Frames, g-orthonormal bases and g-Riesz bases

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 operators is a g-Bessel sequences if and only if the Gram matrix associated to its denes a bounded operator.

Operator-valued bases on Hilbert spaces

In this paper we develop a natural generalization of Schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. We prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. We prove that the operators of a dual ov-basis are continuous. We also dene the concepts of Bessel, Hilbert ov-basis and obta...

New characterizations of fusion bases and Riesz fusion bases in Hilbert spaces

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 denition of fusion dual sequence associated with a fusion basis and show that the operators of a fusion dual sequence are continuous projections. Next we dene the fusion biorthogonal sequence, Bessel fusion basis, Hil...

Generalized Frames in Hilbert Spaces

New Bases for Polynomial-Based Spaces

Since it is well-known that the Vandermonde matrix is ill-conditioned, while the interpolation itself is not unstable in function space, this paper surveys the choices of other new bases. These bases are data-dependent and are categorized into discretely l2-orthonormal and continuously L2-orthonormal bases. The first one construct a unitary Gramian matrix in the space l2(X) while the late...