Introduction to Finite Frame Theory - Frame Research Center

Introduction to Finite Frame Theory 15
Before presenting some insightful basic results in frame theory, we now first
discuss some examples of frames to build up intuition.
3.2 Examples
By Lemma 2 (iii), orthonormal bases are unit-norm Parseval frames (and vice versa).
However, applications typically require redundant Parseval frames. One basic way
to approach this construction problem is to build redundant Parseval frames using
orthonormal bases, and we will present several examples in the sequel. Since the
associated proofs are straightforward, we leave them to the interested reader.
Example 1. Let (e i ) N i=1 be an orthonormal basis for H N .
(1) The system
(e 1 ,0,e 2 ,0,...,e N ,0)
is a Parseval frame for H N . This example indicates that a Parseval frame can
indeed contain zero vectors.
(2) The system
(
e 1 , e 2
√ , e 2
√ , e 3
√ , e 3
√ , e 3
√ ,..., e N
√ ,..., e N
√
),
2 2 3 3 3 N N
is a Parseval frame for H N . This example indicates two important issues: Firstly,
a Parseval frame can have multiple copies of a single vector. Secondly, the norms
of vectors of an (infinite) Parseval frame can converge to zero.
We next consider a series of examples of non-Parseval frames.
Example 2. Let (e i ) N i=1 is an orthonormal basis for H N .
(1) The system
(e 1 ,e 1 ,...,e 1 ,e 2 ,e 3 ,...,e N )
with the vector e 1 appearing N + 1 times, is a frame for H N with frame bounds
1 and N + 1.
(2) The system
(e 1 ,e 1 ,e 2 ,e 2 ,e 3 ,e 3 ,...,e N )
is a 2-tight frame for H N .
(3) The union of L orthonormal bases of H N is a unit-norm L-tight frame for H N ,
generalizing (2).
A particularly interesting example is the smallest truly redundant Parseval frame
for R 2 , which is typically coined Mercedes-Benz frame. The reason for this naming
becomes evident in Figure 1.