G‐Bases in Hilbert Spaces

Xunxiang Guo

doi:10.1155/2012/923729

Abstract

The concept of g‐basis in Hilbert spaces is introduced, which generalizes Schauder basis in Hilbert spaces. Some results about g‐bases are proved. In particular, we characterize the g‐bases and g‐orthonormal bases. And the dual g‐bases are also discussed. We also consider the equivalent relations of g‐bases and g‐orthonormal bases. And the property of g‐minimal of g‐bases is studied as well. Our results show that, in some cases, g‐bases share many useful properties of Schauder bases in Hilbert spaces.

Highlights

In 1946, Gabor 1 introduced a fundamental approach to signal decomposition in terms of elementary signals
It is well known that frames are generalizations of bases in Hilbert spaces
Γj ηj for each j ∈ N, which implies that the dual g-frame of {Λj }j∈N is unique

Summary

Introduction

In 1946, Gabor 1 introduced a fundamental approach to signal decomposition in terms of elementary signals. The following are the standard definitions on frames in Hilbert spaces. A sequence {fi}i∈N of elements of a Hilbert space H is called a frame for H if there are constants A, B > 0 so that. Abstract and Applied Analysis to {Hi : i ∈ N}, which is a sequence of closed subspaces of a Hilbert space V , if there exist two positive constants A and B such that for any f ∈ H. G-frames in Hilbert spaces have been studied intensively; for more details see 12–17 and the references therein. It is well known that frames are generalizations of bases in Hilbert spaces. It is natural to view g-frames as generalizations of the so-called g-bases in Hilbert spaces, which will be defined . The sequence of {Hj : j ∈ N} always means a sequence of closed subspace of some Hilbert space V

Definitions and Lemmas

Characterizations of g-Bases

Equivalent Relations of g-Bases

The Property of g-Minimal of g-Bases