Approximate Identities
We’ve reached the midpoint of the semester in the Fourier class: the mid term is next week. I have free choice in taking it or not, by the simple expedient of showing up or not. I haven’t decided if I want to; on the one hand, I’d have a more meaningful assessment of my progress than the homework, but on the other, I’d have to study :). I’ve gotten accustomed to not doing any studying…
Here’s an interesting concept: approximate identities. Those have been used ingeniously in class to avoid explicitly proving many convergences, ones that we spent so much effort on in my Real Analysis class. The concept we’re using is a little different (less abstract, I suppose, but otherwise conformal to the standard definition, I believe) since they are naturally defined in general Banach algebras, but we’re only dealing with the particular Banach algebra 
. Here’s our definition:
A family
of elements in
is called an approximate identity for
if
for every
The usefulness of an approximate identity is that it is guaranteed— by basic calculus— that 
as 
, so as a result, you have pointwise convergence of the sequence 
to 
almost everywhere.
So far, that has trivialized the proof of the uniqueness of the Fourier transform in , and the inversion theorem, (since the Fejer kernel— not quite the same as the one linked to, but the gist is the same— is an approximate identity) which I’m all for. They also seem suspiciously like the delta function ‘formalisms’ that EE students are introduced to: a series of functions with compact support that shrinks down while the area under the functions remain constant. I suppose this is to be expected, since the ‘real’ identity in the algebra would be the delta function, since it satisfies 
for all functions … except the delta function is not really a function. So we approximate it instead.
Possibly relevant posts:
- simple signal compression (9/28/2005)
- Two ways of doing cool stuff with matrices (2/27/2007)
- Optimization problems (1/24/2006)