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
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
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