Fourier analysis
Decomposing a function into sinusoids, and moving freely between that representation and the original. The corpus holds one text, thibos-fourier-analysis, which teaches it as a working method for experimentalists rather than as a branch of analysis.
The two domains
The subject’s whole content is that a signal has two equivalent descriptions, one a function of time or space and one a set of coefficients on frequencies, and that operations awkward in one are easy in the other. Thibos builds this in order: preliminaries (vectors, phasors, complex numbers, Euler’s formula), then discrete Fourier analysis, then continuous functions with inner products and orthogonality, then the continuous case, sampling theory, the Fourier transform and its properties.
Orthogonality is the load-bearing idea. Sinusoids at different frequencies have zero inner product, so each coefficient can be extracted independently, which is what makes the decomposition unique and computable rather than a fitting problem.
Sampling theory is the second pillar, and the one that connects the mathematics to every device that records anything: what a finite, discretely sampled record can and cannot tell you about the continuous thing behind it.
The statistical chapters
Three chapters set this text apart from a mathematics treatment, and they are the reason it was parked as a methods source rather than a pure one: statistics of Fourier coefficients, hypothesis testing for Fourier coefficients, and directional data analysis with a Rayleigh Z-statistic appendix. A coefficient measured from real data is an estimate with a distribution, and asking whether a spectral peak is real is a statistical question, not a transform one. Most Fourier texts skip this entirely.
Linear systems
The last chapter treats image formation in optics as a linear system, which is the general form the subject takes in application: a system characterized by its response to each frequency, with the output the input’s spectrum multiplied by that response. Filters, lenses and loudspeakers are all described this way.
Where this is used across the hub
Fourier analysis was parked twice on the argument that it is infrastructure beneath at least four spokes and the subject of none. Those four:
../music-tech-wiki— the FFT under every filter, delay, EQ and spectrum analyser.../loudspeaker-design-wiki— frequency-response, impedance and phase measurement.../embedded-iot-wiki— on-device DSP.../psychology-wiki— psychophysics, which is this book’s actual audience. That spoke owns vision-science findings; this owns the transform.
Related
thibos-fourier-analysis · mathematics · probability-theory · vector-calculus · synthesis