This post will answer some of the questions posed at the end of the previous post, about what is a proper definition of sampling sequence.
Let us consider a function space . We are interested in sequences where we can sample a function . The first natural requirement is that the values of the function at uniquely define as an element of , meaning that no other function in can have those same values at . We will call such a sequence a uniqueness sequence.
Definition 1: Uniqueness Sequence
Given a function space , we say that a sequence is a uniqueness sequence if there are no two different functions that agree in .
A uniqueness sequence is not necessarily useful in practice because no continuity (stability) is required, meaning that arbitrarily small errors in the samples can correspond to arbitrarily large errors in the function sampled.
For this reason we impose a stability condition on the definition of sampling.
Definition 2: Sampling Sequence
Given a function space , we say that a sequence is a sampling sequence if there is a positive constant , such that the following inequalty holds for every :
We note that we are imposing that the operator that, given the values of at the sampling sequence, gives is continuous (bounded).
Another important definition is the one of Interpolation Sequence, it can be seen as a dual definition of the one of sampling sequence.
Definition 3: Interpolation Sequence
Given a function space , we say that a sequence is a interpolation sequence if for every square summable sequence there exists a function such that, for every : .
Let us now particularise and consider that is the space of bandlimited function defined in the previous post.
The Fourier Transform is a natural bijection between and . Suppose that is a uniqueness sequence.
We have that, for :
Using the inversion formula one can check that and that all the function in are of this form. If is a non-zero function then is a non-zero function so has to be non-zero for some , then is non-zero for some , so is a complete set in (meaning that it’s span is dense in ). It is straight forward to prove the converse of this statement. This means that uniqueness sequences in correspond to complete sets in .
As we will see sampling sequences are tighly connected to Frames (in the same way uniqueness is connected to complete sets). A further discussion on the topic of Frames will be given on a future post.