CHAPTER 1

Basic Theory for Frames

We begin by giving an elementary self-contained exposition of frames suitable

for our work in subsequent sections. Some of what we describe in this chapter is

known and is standard in the literature. However, our dilation results and exposi-

tion characterizing tight frames and general frames as precisely the direct summands

of Riesz bases seems to be new and serves to clarify some aspects of frame theory

from a functional analysis point of view. For an operator theorist and also for a

Banach space theorist, dilation may be the most natural point of view to take in

regard to frames.

1.1. A Dilation Viewpoint on Frames

Let H be a separable complex Hilbert space. Let B(H) denote the algebra of

all bounded linear operators on H. Let N denote the natural numbers, and Z the

integers. We will use J to denote a generic countable ( or finite ) index set such as

Z, N, Z^2\ NUN (i.e. the disjoint union of two copies of N) etc. The following are

standard definitions:

A sequence {XJ : j G N} of vectors in H is called a frame if there are constants

A, B 0 such that

AWxW'K^Kx^x^^BWxW2

(1)

j

for all x G H. The optimal constants (maximal for A and minimal for B) are

called the frame bounds. The frame {XJ} is called a tight frame if A = B, and is

called normalized if A = B — 1. A sequence {XJ} is called a Riesz basis if it is a

frame and is also a basis for H in the sense that for each x G H there is a unique

sequence {aj} in C such that x — J2

ajxj w

^ h the convergence being in norm. We

note that a Riesz basis is sometimes defined to be a basis which is obtained from

an orthonormal basis by applying a bounded linear invertible operator (cf. [Yo]).

This is equivalent to our definition (cf. Proposition 1.5). Also, it should be noted

that in Hilbert spaces it is well known that Riesz bases are precisely the bounded

unconditional bases.

It is clear from the absolute summation in (1) that the concept of frame ( and

Riesz basis) makes sense for any countable subset of H and does not depend on a

sequential order. Thus there will be no confusion in discussing a frame , or Riesz

basis, indexed by a countable set J.

Prom the definition, a set {XJ : j G J} is a normalized tight frame if and only

if

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