Intuition for Integral Transforms

The Fourier and Laplace transforms are defined by testing the given function f by special functions (characters in the case of Fourier, exponentials in the case of Laplace).

These special functions happen to be eigenfunctions of translation: if one translates a character or an exponential, one gets a scalar multiple of that character or an exponential.

As a consequence, the Fourier or Laplace transforms diagonalise the translation operation (formally, at least).

Whenever two linear operations commute, they are simultaneously diagonalisable (in principle, at least). As such, one expects the Fourier or Laplace transforms to also diagonalise other linear, translation-invariant operations.

Differentiation and integration are linear, translation-invariant operations. This is why they are diagonalised by the Fourier and Laplace transforms.

Diagonalisation is an extremely useful tool; it reduces the non-abelian world of operators and matrices to the abelian world of scalars.

It might help you to think about a discrete model: consider complex valued functions on $Z/n$. The discrete Fourier transform takes $f(k)$ to $g(j) :=\sum_{k=1}^n f(k) \zeta^{jk}$ where $\zeta=e^{2 \pi i/n}$. It is pretty easy to see that, if we change $f(k)$ to $f(k+1)$, we change $g(j)$ to $g(j)*\zeta^j$.

Similarly, changing $f(k)$ to $f(k+1)-f(k)$ changes $g(j)$ to $g(j)*(\zeta^j-1)$. So, in this discrete model, taking a difference becomes multiplication by $(\zeta^j-1)$. In a similar way, in the continuous setting, taking a derivative becomes multiplication by $x$.

You can think of integral transforms as a change of coordinates. One of the key tricks in physics is to pick a coordinate system that makes your problem simpler. For example, you may set your coordinates so that the action you're interested in happens along an axis.

You could think of a Fourier transform as a rotation in a function space. Differentiation is particularly simple in the rotated coordinate system, just as forces are simpler when the coordinate system lines up with the force.

Fourier transform really is a rotation of sorts (an "orthogonal transformation"). If you apply Fourier transform four times, you get back your original function, just as you get sine back when you differentiate four times.