> Actually I thought it is the other way round: A frequency is defined as a sine basically, it has an * amplitude * frequency * phase shift
But this is also true of any periodic wave. The beginner question is, why don't we use triangle waves of square waves? Those also have amplitude, frequency and phase. Frequency just means the reciprocal of the period.
To which my answer is that the magical property of complex exponential functions is that they can be shifted by constant (pointwise) multiplication. Which is a really non-obvious fact at first but is crucial in the machinery.
The complex exponentials constitute an orthogonal basis which diagonalizes the convolution.
I think it's that exponentials simultaneously diagonalize time shifts, derivatives, and integrals. Convolution follows from these -- though there's something to be said for putting convolution above derivatives and integrals in importance.
(A deeper thing going on is that a periodic function can be thought of as a function whose domain is a circle. Circles have obvious rotational symmetry, and when you have symmetry you can use representation theory to decompose things into an (orthogonal) basis. In this case, rotations commute with each other so by some theory the decomposition is going to be entirely through eigenvectors of rotation, which happen to precisely be the exponential functions e^(n theta i) for n an integer. This decomposition is also an isomorphism that carries convolutions to point-wise products in both directions. Also: if you make it so the circle is the complex unit circle, a Fourier transform is the idea that you can create a Laurent polynomial that extends the function to the complex plane minus the origin.)
Nice, indeed at the deeper level it's that if you want to make time shifts equivalent to rotation, you need to move in a circular fashion.
Minor point: to me derivatives are just one specific linear time invariant operator, a kind of convolution (with a generalized function) so I think LTI is the thing we really care about.
Does the representation theoretic perspective / harmonic analysis also explain why the Laplace transform works? I would be interested to know if it's possible to pick a different compact group from S1 to recover the Laplace transform
I've been thinking about this since my original comment actually, but there are complexities from using the noncompact group R. The exponential functions are still the simultaneous eigenvectors of time shifts.
The Laplace transform seems to be just using the fact that <f,g> = integrate(f(x) g(x), x from 0 to infinity) is an inner product for the space of square integrable functions (probably better would be <f,g> = integrate(f(x) conj(g(x)), x from 0 to infinity) as a Hermitian product). The various e^(ax) functions are linearly independent, so the functions g |-> <e^(ax), g> are linearly independent functionals. If the exponential functions are actually enough, then this means you can study a function by studying the vector consisting of its value through all the functionals, which is the Laplace transform.
The Laplace transform has a pretty bad inverse formula, partly because the exponential functions are not orthogonal with respect to the inner product.
Sines and cosines really are fundamental. Pass a triangular, sawtooth or square wave through a narrow-band bandpass filter (an actual physical filter) centred on the fundamental frequency and you'll get a sine wave out the other end. Anything that isn't sinusoidal has components that aren't at the fundamental frequency.
> The beginner question is, why don't we use triangle waves of square waves
You can think in terms of complex exponentials, and the use of complex numbers makes even more sense when you know the differential equations can be solved by polynomial methods, which naturally leads to complex numbers.
However you can also answer "why sine waves not triangle/square waves" with a geometrical answer. (This is how I learned it at school, before I knew about complex numbers or differential equations.)
A property of sine waves is that their sums and products are also sine waves or simple combinations of a small number of sine waves.
The sine wave shape persists. This neat property is unique to sine waves, and you can think of it as a type of symmetry.
For example, in the simplest cases: adding two sine waves with the same frequency and different phase produces another sine wave with that same frequency. Multiplying two sine waves with different frequencies produces the same as a sum of two sine waves, having the sum-of-frequencies and difference-of-frequencies.
Doing so with any other wave shape results in a different wave shape than you started with.
A lot of audio and radio signal processing depends on this property, and the way sum-of-frequencies and difference-of-frequencies works for every sine wave component at the same time. Our whole concept of a radio "band" of signals that you can tune into comes from it.
But this is also true of any periodic wave. The beginner question is, why don't we use triangle waves of square waves? Those also have amplitude, frequency and phase. Frequency just means the reciprocal of the period.
To which my answer is that the magical property of complex exponential functions is that they can be shifted by constant (pointwise) multiplication. Which is a really non-obvious fact at first but is crucial in the machinery.
The complex exponentials constitute an orthogonal basis which diagonalizes the convolution.