Hilbert 空间中的广义正交基

广义正交基是Hilbert 空间中正交基的一个自然推广. 本文首先给出一个广义正交基存在的较弱的充要条件; 然后研究广义正交基的性质, 特别地, 得到广义正交基版本的一些有关正交基的经典性质, 如广义正交基的Bessel 等式和不等式等. 作为广义正交基的一个应用, 本文给出广义Riesz 基的一些新刻画. 最后本文讨论广义框架的冗余问题.

g-Orthonormal bases in Hilbert spaces

g-Orthonormal basis is a natural generalization of orthonormal basis in Hilbert spaces. In this paper, firstly, we give a general condition such that the g-orthonormal bases exist. Then we consider the properties of g-orthonormal basis. In particular, we establish the g-orthonormal basis versions of some classical properties of orthonormal basis, such as Bessel inequality and Bessel equality etc. We also give some new characterizations of g-Riesz bases. Finally, we discuss the redundancy problem of g-frames.