# frame

## Introduction

The concept of a frame is a generalization of the concept of an orthonormal basis: each vector in the space can be represented as a sum of the elements in the frame, but not necessarily uniquely. It is because of this redundancy in representation that frames have found important applications. Surpisingly, despite the fact that frames do not in general consist of orthonormal vectors, the frame representation of a vector may still satisfy the Parseval equation.

## Definition

A frame in a Hilbert space $H$ is a collection $(x_{i})_{i=1}^{n}$ of vectors in $H$ such that there exist constants $A,B>0$ such that for all $x\in H$,

 $A\|x\|^{2}\leq\sum_{i}|\langle x,x_{i}\rangle|^{2}\leq B\|x\|^{2}$

The constants $A,B$ are called the lower and upper frame bounds respectively, or the frame constants. The optimal frame constants $A^{\prime}$ and $B^{\prime}$ are defined respectively as the supremum and infimum of all possible lower and upper frame bounds. When $A=B$, the frame is a tight frame, or an $A$-tight frame. A $1$-tight frame is referred to as a Parseval frame, and the Parseval equation holds with respect to the frame elements:

 $x=\sum_{i=1}^{n}\langle x,x_{i}\rangle x_{i}$

for all $x\in H$.

## Associated Operators (for finite frames)

Let $\dim H=n$ and $\{x_{i}\}_{i=1}^{k}$ be a frame in $H$ ($k\geq n$).

The analysis operator $\theta:H\rightarrow{\mathbb{C}}^{k}$ is the function defined such that $\theta:x\mapsto(\langle x,x_{i}\rangle)_{i}.$

The synthesis operator $\tau:{\mathbb{C}}^{k}\rightarrow H$ is the function defined such that $(c_{i})_{i}^{k}\mapsto\sum_{i=1}^{k}c_{i}x_{i}.$

$\tau=\theta^{\ast}$, that is, $\langle\theta x,y\rangle_{{\mathbb{C}}^{k}}=\langle x,\tau y\rangle_{H}$ for all $x\in H$ and all $y\in{\mathbb{C}}^{k}$.

 $\displaystyle\langle\theta x,y\rangle_{{\mathbb{C}}^{k}}$ $\displaystyle=$ $\displaystyle\langle(\langle x,x_{i}\rangle_{H})_{i=1}^{k},(y_{i})_{i=1}^{k}% \rangle_{{\mathbb{C}}^{k}}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{k}\langle x,x_{i}\rangle_{H}\overline{y_{i}}=\sum_{i=% 1}^{k}\langle x,y_{i}x_{i}\rangle_{H}$ $\displaystyle=$ $\displaystyle\langle x,\sum_{i=1}^{k}y_{i}x_{i}\rangle_{H}=\langle x,\tau y% \rangle_{H}$

The frame operator is defined as the $n\times n$ matrix $\theta^{\ast}\theta:H\rightarrow H$ such that $\theta^{\ast}\theta:x\mapsto\sum_{i=1}^{k}\langle x,x_{i}\rangle x_{i}$ for all $x\in H$.

The Grammian operator is defined as the composition $\theta\theta^{\ast}:{\mathbb{C}}^{k}\rightarrow{\mathbb{C}}^{k}$.

The Grammian and frame operators have the same nonzero eigenvalues.

 Title frame Canonical name Frame1 Date of creation 2013-03-22 14:25:37 Last modified on 2013-03-22 14:25:37 Owner swiftset (1337) Last modified by swiftset (1337) Numerical id 9 Author swiftset (1337) Entry type Definition Classification msc 46C99 Related topic RieszSequence Related topic SetOfSampling Defines Parseval frame Defines tight frame Defines frame constants Defines frame bounds Defines synthesis operator Defines analysis operator Defines frame operator Defines Grammian