From Weyl-Heisenberg Frames to Infinite Quadratic Forms

Let $a$, $b$ be two fixed positive constants. A function $g\in L^2({\mathbb R})$iscalleda\mathit{\text{mother}}\mathit{\text{Weyl-Heisenberg}}\mathit{\text{frame}}\mathit{\text{wavelet}}for$(a,b)$if$g$generatesaframefor$L^2({\mathbb R})$undermodulatesby$b$ and translates by $a$, i.e., $\{e^{imbt}g(t-na{\}}_{m,n\in \mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. In this paper, we establish a connection between mother Weyl-Heisenberg frame wavelets of certain special forms and certain strongly positive definite quadratic forms of infinite dimension. Some examples of application in matrix algebra are provided.