3 ms·

Back in my uni days I did not get why that works. Why are sine waves special? Turns out... they are not! You can do the same thing using a different set of fun

Back in my uni days I did not get why that works. Why are sine waves special? Turns out... they are not! You can do the same thing using a different set of functions, like Legendre polynomials, or wavelets.

cjbgkagh 1y ago
Another place where functions are approximated is in machine learning which use a variety of non-linear functions for activations, for example the ReLU f(x)= max(0,x)
MontyCarloHall 1y ago
>Turns out... they are not! You can do the same thing using a different set of functions, like Legendre polynomials, or wavelets. Yup, any set of orthogonal functions! The special thing about sines is that they form an exceptionally easy-to-understand orthogonal basis, with a bunch of other nice properties to boot.
nestes 1y ago
To be maximally pedantic, sine waves (or complex exponentials through Euler's formula), ARE special because they're the eigenfunctions of linear time-invariant systems. For anybody reading this without a linear algebra background, this just means using sine waves often makes your math a lot less disgusting when representing a broad class of useful mathematical models. Which to your point: You're absolutely correct that you can use a bunch of different sets of functions for your decomposition. Linear algebra just says that you might as well use the most convenient one!
- MontyCarloHall 1y ago
  >They're eigenfunctions of linear time-invariant systems For someone reading this with only a calculus background, an example of this is that you get back a sine (times a constant) if you differentiate it twice, i.e. d^2/dt^2 sin(nt) = -n^2 sin(nt). Put technically, sines/cosines are eigenfunctions of the second derivative operator. This turns out to be really convenient for a lot of physical problems (e.g. wave/diffusion equations).

3 ms·

Back in my uni days I did not get why that works. Why are sine waves special? Turns out... they are not! You can do the same thing using a different set of fun

by incognito124 1y ago

cjbgkagh 1y ago
Another place where functions are approximated is in machine learning which use a variety of non-linear functions for activations, for example the ReLU f(x)= max(0,x)
MontyCarloHall 1y ago
>Turns out... they are not! You can do the same thing using a different set of functions, like Legendre polynomials, or wavelets. Yup, any set of orthogonal functions! The special thing about sines is that they form an exceptionally easy-to-understand orthogonal basis, with a bunch of other nice properties to boot.
nestes 1y ago
To be maximally pedantic, sine waves (or complex exponentials through Euler's formula), ARE special because they're the eigenfunctions of linear time-invariant systems. For anybody reading this without a linear algebra background, this just means using sine waves often makes your math a lot less disgusting when representing a broad class of useful mathematical models. Which to your point: You're absolutely correct that you can use a bunch of different sets of functions for your decomposition. Linear algebra just says that you might as well use the most convenient one!
- MontyCarloHall 1y ago
  >They're eigenfunctions of linear time-invariant systems For someone reading this with only a calculus background, an example of this is that you get back a sine (times a constant) if you differentiate it twice, i.e. d^2/dt^2 sin(nt) = -n^2 sin(nt). Put technically, sines/cosines are eigenfunctions of the second derivative operator. This turns out to be really convenient for a lot of physical problems (e.g. wave/diffusion equations).