强正定二次型与Weyl-Heisenberg框架的构造

On the strongly positive definite quadratic forms and the construction of Weyl-Heisenberg frames

本文研究了一类具有特殊结构的无限维二次型,得到这类二次型的对称矩阵是符号为多项式的模的平方的Laurent矩阵,进一步得到了这类二次型是强正定的判断标准以及一类Weyl-Heisenberg框架的构造.本文还研究了这类二次型的矩阵的所有有限维主对角子矩阵的强正定性,并由此得到一类子空间Weyl-Heisenberg框架的构造.最后举例说明本文的主要结果及其应用.本文建立了两个看似不相关的领域间的联系.

In this paper,a kind of infinite quadratic forms with special structure are studied.We show that the symmetric matrices of this kind of infinite quadratic forms are Laurent matrices with symbols the square of the module of some polynomails.Then the criterion of the strongly positive definiteness of the infinite quadratic forms and the construction of a kind of Weyl-Heisenberg frames are obtained.We also consider strongly positive definiteness of the diagonal main block sub-matrices of the matrices of the special infinite quadratic forms and the construction of subspace Weyl-Heisenberg frames is established.Finally,some examples are provided for illustrating our main results and their applications.Our results make an interesting connection between two seemingly irrelevant subjects.