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 $x\in H$,

$$A{\parallel x\parallel }^2\le \sum _i{|⟨x,x_i⟩|}^2\le B{\parallel x\parallel }^2$$

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:

$$x=\sum _{i=1}^n⟨x,x_i⟩x_i$$

for all $x\in H$.

Associated Operators (for finite frames)

Let $dimH=n$ and ${\{x_i\}}_{i=1}^k$ be a frame in $H$ ($k\ge n$).

The analysis operator $\theta :H\to {ℂ}^k$ is the function defined such that $\theta :x\mapsto {(⟨x,x_i⟩)}_i.$

The synthesis operator $\tau :{ℂ}^k\to H$ is the function defined such that ${(c_i)}_i^k\mapsto {\sum }_{i=1}^k{c}_i{x}_i.$

$\tau ={\theta }^{\ast }$, that is, ${⟨\theta x,y⟩}_{{ℂ}^k}={⟨x,\tau y⟩}_H$ for all $x\in H$ and all $y\in {ℂ}^k$.

${⟨\theta x,y⟩}_{{ℂ}^k}$ $=$ ${⟨{({⟨x,x_i⟩}_H)}_{i=1}^k,{(y_i)}_{i=1}^k⟩}_{{ℂ}^k}$

$=$ ${\displaystyle \sum _{i=1}^k}{⟨x,x_i⟩}_H\overline{y_i}={\displaystyle \sum _{i=1}^k}{⟨x,y_i{x}_i⟩}_H$

$=$ ${⟨x,{\displaystyle \sum _{i=1}^k}{y}_i{x}_i⟩}_H={⟨x,\tau y⟩}_H$

The frame operator is defined as the $n\times n$ matrix ${\theta }^{\ast }\theta :H\to H$ such that ${\theta }^{\ast }\theta :x\mapsto {\sum }_{i=1}^k⟨x,x_i⟩x_i$ for all $x\in H$.

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

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