Re:Generalizations of Fourier Analysis
by nextaccountic on Sunday April 16, 2023 @11:36PM
(#35596602)
So, so I'm going with a rusty knowledge of a computer engineering course from years ago,
> a map of which "global decay timescale" fits your data best
What do you mean by this?
> -- since each "slice" is itself sufficient to fully describe the time series
But each slice is multiplied by an exponent.. which, okay, becomes a convolution that lets you recover the original function
Like how the DC term of Fourier transform is the baseline average, i.e. global energy constant (per time), the baseline decay describes the global energy dissipation or amplification (per time).
Another useful way of looking at it: Laplace transform is doing many extra Fourier-transform-but-with-decay giving you a map of which "global decay timescale" fits your data best -- since each "slice" is itself sufficient to fully describe the time series
They are all cases of integral transforms with different choices of the set of "primitive fingerprints" -- see chirp transform, wavelet transform, chirplet transform etc -- all taking advantage of the fact that if you choose one set of basis "brushes" that are not redundant with each other (e.g. having red-green-blue brushes is independent, as is magenta-green-yellow but having red-green-blue-yellow is not) then you will be able to describe your signal in terms of a composition of those kernels.