Representing sum of delta functions as sum of exponentials

[RobikShrestha]

I saw this formula while studying Fourier Transform:

[Sum of δ(t-lT) from l=-infinity to l= +infinity] = 1/T * [Sum Of exp(jlΩt) from l=-infinity to l=+infinity].

I am having trouble getting this into my head. How do I prove it or understand the physical meaning of this formula?

 

[Hurkyl]

Physical meaning depends on how you are modeling the physical world with mathematical objects.

Eyeballing it, I think there's a clear geometric idea here, though. The sum of delta functions is periodic, concentrated at one point per period. The sum of exponentials sets up a sequence of waves aligned with that period that constructively interfere at those points and destructively interfere at the remaining points.I feel it important to point out that neither side makes sense as a function -- this equation could only possibly make sense when viewed as a statement about generalized functions.How to prove it? The most direct way would be by the typical definition of generalized functions: multiply by a generic test function and integrate. If both sides give the same value for all test functions, they are equal as generalized functions.

 

[RobikShrestha]

Generalized functions? Generic test function? What are they?

It seems I have a lot of studying to do. They are not taught in our college course.

If you have simple explanations then please explain them to me.

 

[Hurkyl]

The short answer is that generalized function is something you can integrate with, and their meaning is entirely determined by integration.

For example, if F(x) and G(x) are generalized functions, then we can make sense of their sum F(x) + G(x) as a generalized function: it is defined to be the generalized function such that

$$\int (F(x) + G(x)) \varphi(x) \, dx = \int f(x) \varphi(x) \, dx + \int G(x) \varphi(x) \, dx$$

for all test functions $\varphi$.

(aside, you should be wary about asking questions like "what is the value of this generalized function at 3?", which has a lot of subtle semantic difficulties)

What is a test function? There are lots of choices for what to call a test function, and each different choice gives a different class of generalized functions.


Two typical definitions are are

• A test function is a smooth function with compact support (i.e it is zero outside of a bounded set)
• Rapidly decreasing functions: smooth functions that vanish 'rapidly' as $x \mapsto \pm \infty$.

It would probably be fair to equate "test function" with "well-behaved function".


There are other schemes for generalizing the notion of function. The ones I describe here are called distributions.