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

TL;DR

This paper extends frame construction methods to finite-dimensional Pontryagin spaces, demonstrating how frames can be derived from operators and embedded into larger spaces with specific properties.

Contribution

It introduces new techniques for constructing and extending frames in finite-dimensional Pontryagin spaces, including coupling frames from different spaces.

Findings

Any frame in a finite-dimensional Pontryagin space is a $J$-orthogonal projection of a frame in a larger space.

Constructs a Pontryagin space containing given frames as subsets.

Provides methods to build a common Pontryagin space embedding multiple frames.

Abstract

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 $\{k_{n}\}_{n=1}^{m},\{x_{n}\}_{n=1}^{k}\subset \{y_{n}\}_{n=1}^{N}$.