I am new to Wavelet Theory. My mind came across one question. We learn about Fourier Series (FS) and then about Fourier Transform (FT). Then, why are we not dealing with "Wavelet Series" as FS and directly goes to Wavelet Transform?
Is there any reason for this or I am asking weird question?
Since dilated and translated families of wavelets consitute orthogonal basis of $L^2(\mathbb{R})$, you can decompose any square-integrable signal as wavelet series, quite similarly to a Fourier series expansion. So it makes perfect sense to talk about "wavelet series".
Now about the reason that justifies that wavelet series are classicaly introduced after the wavelet transform (note that it is not always the case, for example the wikipedia article introduces wavelet basis before the wavelet transform). I think it is partly motivated by the fact that the main theorems of wavelet basis are easier to prove if you already know the wavelet transform. Look at the 7th chapter of Mallat (2008), and especially at the Mallat-Meyer theorem (theorem 7.3) which states that dilated and translated families of wavelets are indeed orthogonal basis of $L^2(\mathbb{R})$.
Mallat, S. (2008). A wavelet tour of signal processing: the sparse way. Academic press.
https://mathematica.stackexchange.com/questions/263538/coupled-heat-transfer-equations-using-collocation-method/263543?noredirect=1#comment702040_263543
Can it be complete? I am not sure in contrast to the author of that post.
– Igor Kotelnikov Mar 02 '23 at 04:07