The referenced article has: "Wavelets came about as a kind of update to an enormously useful mathematical technique known as the Fourier transform. In 1807, Jo

The referenced article has: "Wavelets came about as a kind of update to an enormously useful mathematical technique known as the Fourier transform. In 1807, Joseph Fourier discovered that any periodic function — an equation whose values repeat cyclically — could be expressed as the sum of trigonometric functions like sine and cosine." Due to the "periodic", that math is Fourier series, not the Fourier transform. Given almost any periodic function, the Fourier transform won't exist. The math of Fourier theory is done carefully in two of the W. Rudin texts: Fourier series is in his Principles of Mathematical Analysis, and Fourier transforms is in his Real and Complex Analysis. For more, his Functional Analysis covers the related distributions. See also the "celebrated" Riesz–Fischer theorem, IIRC in his Real and Complex Analysis. As I recall, wavelets have some good completeness properties. Power spectra are of interest, and for that see the Wiener–Khinchin theorem. For the relevant statistics of estimating power spectra, see the work of Blackman and Tukey. The referenced article also has "That’s because Fourier transforms have a major limitation: They only supply information about the frequencies present in a signal, saying nothing about their timing or quantity." Phase gives some information on timing, and the power spectra, on quantity. The article also has "Whenever you have a particularly good match, a mathematical operation between them known as the dot product becomes zero, or very close to it." A dot product value of zero means orthogonality, that is, in the usual senses, neither signal is useful for saying any much about the other. Of course, in the case of stochastic processes, the orthogonality of a dot product (inner product) is not the same as probabilistic independence. For computations, of course, see the many versions of the fast Fourier transform, in response to a question from R. Garwin, rediscovered by Tukey, first programmed by Cooley, later given various developments and many applications. At one point early in my career, these topics got the company I was working for "sole source" on a US Navy contract and got me a nice, new high end Camaro, a nice violin, a nice piano for my wife, some good French food, a sack full of nice Nikon camera equipment, and a nice stock market account! My annual salary was 6+ the cost of the Camaro.