Fourier decomposition is another way to depict a series as a composition of sinusoidal functions. It's also called a Fourier series. It's actually possible to represent almost any kind of wave as a sum of sines and cosines. One really useful application of this is compression, because if a Fourier series converges, it converges really quickly. This compression is used in MP3 formats as well as JPEGs.
keenan
@panda Yep, great comments. Fourier-type compression starts to get challenging as we move to more complicated domains, like curved surfaces in space. A key problem is that there is no longer an $O(n \log n)$ fast Fourier transform, which means one has to resort to more expensive computational alternatives. Some problems of this type can be addressed via wavelets and subdivision surfaces, though we're still far from a real replacement for the FFT.
Fourier decomposition is another way to depict a series as a composition of sinusoidal functions. It's also called a Fourier series. It's actually possible to represent almost any kind of wave as a sum of sines and cosines. One really useful application of this is compression, because if a Fourier series converges, it converges really quickly. This compression is used in MP3 formats as well as JPEGs.
@panda Yep, great comments. Fourier-type compression starts to get challenging as we move to more complicated domains, like curved surfaces in space. A key problem is that there is no longer an $O(n \log n)$ fast Fourier transform, which means one has to resort to more expensive computational alternatives. Some problems of this type can be addressed via wavelets and subdivision surfaces, though we're still far from a real replacement for the FFT.