Approximating approximations

[K.I.L.E.R]

Given that most of the work I do is pretty much outside the scope of my current course I can't really get an answer on this.
I personally am going to work on this myself however I would like to know what the mathematically oriented people of this forum think about this.

Basically If I solve an Nth order, Rth degree ODE using a Laplace transform and fit the resulting function to a Fourier series and proceed to use a Taylor series to approximate result what would be the stability of such a system?

Obviously the first thing that comes to mind would be that not everything will fit well in this as a non-periodic function will have to be approximated using a Fourier series, which is why I'm calling this an approximation of an approximation.

Another thing came to mind was what if I were to use wavelets in order to offset the problem of the ODE being non-periodic?

Thanks.

 

[Fred]

"Basically If I solve an Nth order, Rth degree ODE using a Laplace transform and fit the resulting function to a Fourier series and proceed to use a Taylor series to approximate result what would be the stability of such a system?"

Well assuming you are using your approximating functions properly (eg they are well defined in your domain and convergent* whenever you apply them) then the only thing you might be destroying is the rate of convergence.

But I don't see why you would want to do that.. Keep in mind the first few orders of a series might be (and usually are) horrible approximations so the zeal to get nice simple functions is usually going to work against you.

Worse, lets say you taylor expanded a generic fourier series. You are going to get first order contributions from infinitely many terms, so you need to be able to resum them somehow to get something that even looks tractable (this usually is NOT possible).

Keep in mind what I said above. You have to make sure your domains *match* the range of the preceding approximation, and you have to worry about different types of convergence.