- #1

solanojedi

- 34

- 0

I'm an electrical engineer trying to make a little bit of order in my mind about eigenfunction theory, sturm liouville operators (regular and singular) on my own free time and I thought I was grasping almost everything, but I had a little doubt that is actually coming from the field of control theory.

First, I start from the math: I learned (I'll be for sure approximate, I apologize, but correct me if I'm wrong) that in Hilbert spaces it is always possible to define a basis made of eigenfunctions and every operator and function existing inside the space can be rewritten using that basis. However, how to find this basis?

There are particular operators called Sturm Liouville operators that, defining an eigenvalue problem, can generate a set of functions and those functions are the basis for the corresponding space I'm working with. In particular, there are Sturm Liouville "singular" problems that are defined over a space that is not "finite" but extends from [itex]-\infty[/itex] to [itex] +\infty [/itex]: one of them, and for an engineer the most important, is [itex] L^2(-\infty, +\infty) [/itex].

In this space, the operator [itex]\frac{d^2}{dx^2}[/itex] defines an Sturm Liouville singular problem and the solution of this eigenvalue problem is an infinite set of complex exponentials. From this theory we get the Fourier transform theory.

Ok, this was a sort of a very general summary of what I have in my mind.

Now, switching to control theory: we say that for LTI systems (that are described with LTI operators) the complex exponential is an eigenfunction of the operator and hence I can use the Fourier transform to analyze those systems.

This, in my opinion, is only because I'm dealing with signals and operators that by definition are [itex] L^2(-\infty, +\infty) [/itex]. Hence, I believe LTI operators are only a special case of the general operators that are in [itex] L^2(-\infty, +\infty) [/itex].

**Here's my question: what happens if the system is not LTI but for example linear time variant? What kind of signals (and operators) I'm dealing with?**I guess that they are not anymore [itex] L^2(-\infty, +\infty) [/itex], and so I cannot use the same instruments like Fourier/Laplace transform , but this is just a guess.

Thank you in advance!

Nicola