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 [13]. we show that a continuous frame is riesz-type if and only if it is a con...

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 [13]. We show that a continuous frame is Riesz-type if and only if it is a con...

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...

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 de ne the concepts of bessel, hilbert ov-basis and obt...

in this paper we investigate a new notion of bases in hilbert spaces and similarto fusion frame theory we introduce fusion bases theory in hilbert spaces. we also introducea new de nition of fusion dual sequence associated with a fusion basis and show that theoperators of a fusion dual sequence are continuous projections. next we de ne the fusionbiorthogonal sequence, bessel fusion basis, hilbe...

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...

An orthonormal basis adaptation method for function approximation was developed and applied to reinforcement learning with multi-dimensional continuous state space. First, a basis used for linear function approximation of a control function is set to an orthonormal basis. Next, basis elements with small activities are replaced with other candidate elements as learning progresses. As this replac...

A provably stable reduced order model, based on a projection onto a scaled orthonormal Laguerre basis, followed by a SVD step, is proposed. The method relies on the conformal mapping properties induced by the complete orthonormal scaled Laguerre basis, allowing a mapping from the discrete-stable case to the continuous-stable case and vice versa.

Square-integrable functions f ∈ L are those of length ‖f‖L2 = 〈f, f〉 L2 < ∞. A subset D ⊂ L is said to be dense, if any f ∈ L can be approximated by a sequence fn ∈ D. This means limn ‖f − fn‖L2 = 0. An orthonormal basis {Ωn} is a set of orthonormal vectors whose finite linear combinations are dense. A linear transformation T is continuous on L if ‖Tf‖L2 ≤ M‖f‖L2 for some constant M < ∞. A cont...