# characterization of tight frames in ${\mathbb{R}}^{n}$

###### Question 1.

What conditions must the vectors $\{x_{i}\}_{i=1}^{k}\subset{\mathbb{R}}^{n}$ satisfy in order to be a tight frame in ${\mathbb{R}}^{n}$?

###### Solution.

Let $E$ be the $k\times n$ matrix whose rows are the vectors $\{x_{i}\}_{i=1}^{k}$:

 $E=\begin{pmatrix}x_{i,1}&\cdots&x_{i,n}\\ \vdots&\ddots&\vdots\\ x_{k,1}&\cdots&x_{k,n}\end{pmatrix}.$

Then the tight frame condition $\sum_{i=1}^{k}|\langle x,x_{i}\rangle|^{2}=A\|x\|^{2}$ gives $(Ex)^{T}Ex=Ax^{T}x$ for all $x\in{\mathbb{R}}^{n}$, or $E^{T}E=AI_{n}$:

 $E^{T}E=\begin{pmatrix}\sum_{i=1}^{k}x_{i,1}x_{i,1}&\cdots&\sum_{i=1}^{k}x_{i,1% }x_{i,k}\\ \vdots&\ddots&\vdots\\ \sum_{i=1}^{k}x_{i,k}x_{i,1}&\cdots&\sum_{i=1}^{k}x_{i,k}x_{i,k}\end{pmatrix}=% \begin{pmatrix}A&\cdots 0\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots 0\cdots&A\end{pmatrix}=AI_{n}.$

Therefore, the vectors

 $\left\{x_{i}=\begin{pmatrix}x_{i,1}\\ \vdots\\ x_{i,n}\end{pmatrix}\right\}_{i=1}^{k}\subset{\mathbb{R}}^{n}$

are an A-tight frame iff the vectors

 $\left\{x^{\prime}_{i}=\begin{pmatrix}x_{1,i}\\ \vdots\\ x_{k,i}\end{pmatrix}\right\}_{i=1}^{n}\subset{\mathbb{R}}^{k},$

i.e., the columns of $E$, are all of norm $\sqrt{A}$ and form an orthogonal family.

Title characterization of tight frames in ${\mathbb{R}}^{n}$ CharacterizationOfTightFramesInmathbbRn 2013-03-22 14:27:08 2013-03-22 14:27:08 swiftset (1337) swiftset (1337) 5 swiftset (1337) Result msc 46C99