A reading approach to time-frequency analysis

0) Functional Analysis [Werner]
I numbered this section zero because this is absolutely a must. If you haven’t dealt with functional analysis (FA) recently then you want to recall its central topics. You have to be familiar with all those properties of normed spaces, Banach spaces (, , etc.) and Hilbert spaces. You should know the connection between the Fourier transform (FT) and Sobolev spaces and understand why the FT extends to as a unitary operator. You might spend some time on this. Personally, I chose [Werner], a German book about FA, starting with normed spaces and looking at adjoint operators before discussing Hilbert spaces - The whole chapter V is of high interest, especially V.2 about FT and Sobolev spaces. Here, you already get familiar with all those core properties of the FT, and you will calmly go through the first chapters of

1a) Foundations of Time-Frequency Analysis [Gröchenig]
You should really understand what Gröchenig is talking about and have already seen proofs in [Werner] for some of the properties he mentions. You should be able to follow his proofs at every step, e.g., because you already know why .

At least for the start, you can read this book independently from the next two I mention. Of course you’ll notice similarities to the Gabor frame chapters in [Christensen]. Reaching chapter 9 in [Gröchenig], I admit to have only read 9.1 and skipped representation theory. Also, the generalization to pseudodifferential operators might not be of such interest.

1b) Wavelets [Blatter]
After FA, you can start with [Blatter] at the same time as with [Gröchenig], therefore I numbered this also as 1. In the first chapters, it provides an easy-to-read approach into the concepts of the analysis and synthesis of functions, Fourier series and FT. Being familiar with FA, you will calmly confirm the content. You will notice the affinity between the properties of the STFT (which you already came over in [Gröchenig]) and those of the Wavelet transform. It is a rather small book, so you will proceed quite quickly. You might want to skip some technical proofs and the last chapter about Wavelet constructions. During [Blatter], you will read about Shannon’s sampling theorem, the uncertainty principle (which you also already know from [Gröchenig]) and frames. But before reading its chapter 4 about frames, be sure to have started with

2) Frames and Riesz Bases [Christensen]
[Blatter] uses other notations as [Christensen], where of course you should grant [Christensen] more weight. What [Blatter] already calls the frame operator is just the adjoint of the pre-frame operator (i.e., the analysis operator ) in [Christensen], and [Blatter]’s Gram-matrix is now [Christensen]’s frame operator ; [Christensen]’s Gram-matrix is then adjoint to [Blatter]’s, but both are written as . But nevertheless, you will soon identify the theorems both books discuss. Be careful, what [Christensen] calls is not just a set, but a sequence.

The connection to TF-analysis raises when Gabor frames are examined. You will also step over these in [Gröchenig]. You might want to leave out the Wavelet chapters here, as [Blatter] already showed you enough if you’re just interested in Gabor analysis. Again, the representations of locally compact groups might not be of that interest.

3) Gabor Analysis [Feichtinger/Strohmer]
I noticed that it was of great worth to have a good preparation before starting with this book. The first three pages of chapter 1 already cite many of the properties of (Gabor) frames you should already have seen in detail. At this very beginning I noticed references to the chapters 5 in both [Gröchenig] and [Christensen]. Just for the start, the applied chapters may not be of such interest as the theory chapters in the first half of the book. But note that this book is not to be read linearly. E.g. for chapter 1, it is enough to make references to theorems of the above books, and for chapter 3 about Feichtinger’s algebra I recommend to already have read the chapters 11+12 of [Gröchenig].

Besides those books, there are some further sources like papers and theses which themselves give hints to more sources, discussing further topics like Banach Gelfand triples, Wiener amalgam spaces, and more. At this point, you have already finished starting and are now already progressing, where I don’t think I have to guide you further.

Literature:
[Blatter] C. Blatter: Wavelets - Eine Einführung, Vieweg
[Christensen] O. Christensen: An Introduction to Frames and Riesz Bases, Birkhäuser
[Feichtinger/Strohmer] H. G. Feichtinger, T. Strohmer: Gabor Analysis and Algorithms - Theory and Applications, Birkhäuser
[Gröchenig] K. Gröchenig: Foundations of Time-Frequency Analysis, Birkhäuser
[Werner] D. Werner: Funktionalanalysis, Springer