Construction, Extension and Coupling of Frames on Finite Dimensional Pontryagin Space

In this paper we extend to finite-dimensional Pontryagin spaces the methods used in \cite{CasazzaLeon,Deguang} to build frames from an adjoint and positive operator. It is proved that any frame in finite dimensional Pontryagin space $\mathcal{K}$ is $J$-orthogonal projection of a frame for a space $\mathcal{H}$ such that $\mathcal{K}\subset \mathcal{H}$. Furthermore, given $\{k_n{\}}_{n=1}^m$ and $\{x_n{\}}_{n=1}^k$ frames for $\mathcal{K}$ and $\mathcal{H}$ respectively, we build a finite-dimensional Pontryagin space $\mathfrak{K}$ and a frame $\{y_n{\}}_{n=1}^N$ for $\mathfrak{K}$ such that $\mathcal{K},\mathcal{H}\subset \mathfrak{K}$ and $\{k_n{\}}_{n=1}^m,\{x_n{\}}_{n=1}^k\subset \{y_n{\}}_{n=1}^N$.