[maliker]

Yes.

A mathematician explained it to me as an alternative to a fourier transform: instead of describing the function as a sum of sine waves, its a sum of mexican hats (or whatever basis function). And it turns out that’s a simpler representation in the case of function with sharp discontinuities. It’s also an alternative to a Taylor series, replacing sums of derivatives with the sum of the scaled basis function. Seemed like a pretty elegant explanation to me.

[a-dub]

yes, this is how i understand them.

it has been a long time, so i'm probably going to say something really stupid... but it's fun to try and hopefully someone will correct me if i'm wrong,

my understanding is that one place that they're useful is in terms of time-frequency uncertainty that occurs with regular fourier transforms. for example, if you're doing spectral analysis and looking for low frequencies you need a wider analysis window (a full cycle has a long period for low frequencies), but that wide analysis window now covers a lot of time, so if you pick up a low frequency component, you don't know where it is in that big wide window. wavelets instead fit these families of arbitrary basis functions that can be more compact, so you can use a narrower window and therefore have less time uncertainty for the things you find within it.

...most applications sort of go from there, where you want to warp the equivalent of the x-axis in the frequency domain in various ways and perhaps be able to either use a narrower analysis window for better time resolution or say, a smaller number of basis functions for reconstruction in lossy compression where some bands are less important than others and resolution in those bands can be discarded.

[mattkrause]

I think about it in terms of trading in time-frequency space. The total amount of time-frequency uncertainty is conserved: it's just like the Heisenberg uncertainty principle from physics, and is in fact deeply connected to it.

A classic Fourier spectrogram makes one choice about time uncertainty vs. frequency uncertainty and uses it for all frequencies. You can think of it as dividing time-frequency space into little squares that work okay for most quantities of interest.

Wavelets alter that tradeoff. At low frequencies, we often care about small differences in frequency (e.g., 4 vs 5 Hz), while the precise location of the peaks (e.g., 10 vs 10.1 s) matters less because the signal is changing slowly anyway. The situation is reversed at high frequencies: 1 kHz vs 1.001 kHz is often physically irrelevant, but the timing matters because they're so short). We therefore divide the time-frequency space up so that the windows are long in time, but narrow in frequency at low frequencies, square in the middle, and short in time but wide in frequency at higher ones.

On top of that, wavelets recognize that sine waves are a mathematically convenient basis, but may not reflect your data, so you can also swap in a "mother wavelet" that better matches whatever phenomena you're studying.