Question

What is the basis for a Laplace transform? In other words, what is the fundamental reasoning behind such a transform?

Answer 1

I don't remember the specific steps to do so, but I remember the premise was that certain differential functions are very difficult to solve so we 'translate' it into another domain (laplace domain), solve it there, and then do an inverse laplace transform to get back to the original.

ah memories... *shivers*

Answer 2

I've usually seen it described as a transformation from a time domain to a frequency domain. It's a projection of a function onto an orthogonal basis (in this case one dimensional).

For more information look at integral transforms in general, as well as the idea of inner products from linear algebra.

Answer 3

it's similar to how an discrete FFT works (although very unrelated to Laplace): we translate a function into a pointwise vector, do our work on it, then interpolate it back to an actual function.

full disclosure: I may be completely wrong.

Answer 4

polpak, i think you are referring to FFT, no? an FFT involves translating a signal from time domain to freq domain.

Answer 5

Nope, I mean Laplace.

Answer 6

" Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time."

Answer 7

very interesting. apparently the two (fft, laplace) have very similar uses.

...brain is groooowing.

Answer 8

Indeed. And the principles behind them are also similar (projections onto orthogonal bases).

Answer 9

I got aloooott of learning ahead of me it seems ;) and who knew Matt was such the genius? kudos ...