Fourier analysis textbook What would be a good textbook for learning Fourier analysis if one is comfortable with Measure Theory, Hilbert spaces and $L^p$ spaces in general? Other sources like video lectures would also be helpful if available. I am not familiar with Complex Analysis.
 A: One of the best introductory Fourier analysis textbook in my eyes is


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*Fourier analysis by J. Duoandikoetxea


The author explains things extremely well and chooses just the right level of detail. However, there are unfortunately no exercises.
Another fairly recent one is


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*Classical and multilinear harmonic analysis by W. Schlag and C. Muscalu


This work consists of two volumes, the first deals with introductory classical Fourier analysis material, while the second is more modern and touches on ongoing research. It has lots of exercises.
The book by Katznelson mentioned in the comment I can also recommend. It focuses on Fourier series and the exercises are nice.
Then of course the classical texts by E. M. Stein are a must-read:


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*Introduction to Fourier analysis on Euclidean spaces

*Singular integrals and differentiability properties of functions

*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals
They should be read in that order. The second and third are already quite advanced and may be a bit hard to digest for a beginner since the presentation is quite dense and fairly general. The third book in particular is considered by many to be the standard reference regarding many of the core topics in harmonic analysis.
To complement the other answer, here is my opinion on the Grafakos texts: while the books are very well and thoroughly written, they are arguably not a good single source to learn the subject. The focus is on formalities, precision and long-winded detailed calculations instead of ideas and concepts. The books are a good reference because they contain extremely detailed proofs, cover a lot of material and contain quite general statements of important results. There are also lots of exercises. So maybe use these alongside a more conceptual textbook to look up details you don't understand in the other book or to get some practice from doing the exercises. It's always good to have multiple sources anyways, because different authors often explain the same thing in slightly different ways.
A: I would recommend the book Classical Fourier Analysis by Loukas Grafakos published by Springer. It is a vast, extensive and highly formal introduction to the subject which suffices for most purposes. However, it is one of the hardest analysis books I know, but this can also originate from the subject itself. One big plus of this book is the large amount of exercises provided.
