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There are three of them. 1. Fourier transform: unbounded real domain <-> unbounded real domain 2. Fourier series: bounded real domain <-> unbounded integer do
by leni536 10y ago
There are three of them.
1. Fourier transform: unbounded real domain <-> unbounded real domain
2. Fourier series: bounded real domain <-> unbounded integer domain
3. Discrete Fourier transform (DFT): bounded integer domain <-> bounded integer domain
FFT is an algorithm for quick calculation of DFT, which is not Fourier series.
- enriquto 10y agoyou are missing the discrete time fourier transform, which is the dual of your point (2). Together, these eight domains can be arranged on a commutative cube, all of whose edges have beautiful meanings: sampling, interpolation, etc. The fact that the cube is commutative gives a lot of theorems relating each transform and these operations.
- deleted 10y ago[deleted]
- dnautics 10y agoThe point being that by your definition of not, DFT is also NOT a Fourier transform. It's only a "transform" in that the two domains are similar (identical, mod complexity). That's quite confusing to engineers who only care about the results. Computationally, the first two are strictly speaking impossible, but effectively, your data come in being modeled as finite real-valued arrays that represent function values, (close enough to "bounded real domain") and need to come out as finite real-valued arrays, represented integer-indexed frequencies, and a DFT on your data will operationally give you that. This is much more realistically described as a discretized pseudo-fourier series.