Parseval frames with n+1 vectors in R^n

TL;DR

This paper constructs specific Parseval frames with n+1 vectors in n-dimensional space containing a given vector and characterizes when unit-norm frames can be scaled to form Parseval frames.

Contribution

It provides a method to construct Parseval frames with n+1 vectors containing a specified vector and characterizes scalable unit-norm frames.

Findings

Constructed Parseval frames with n+1 vectors in R^n containing a given vector.

Characterized when unit-norm frames can be scaled to Parseval frames.

Abstract

We construct a Parseval frame with $n+1$ vectors in $\R^n$ that contains a given vector. We also provide a characterization of unit-norm frames that can be scaled to a Parseval frame.