Books for beginners on the Fourier Transform

One of the most elementary yet rigorous introduction to Fourier analysis is, in my opinion, the textbook [1] by Baggett and Fulks. Its strong points are the following ones:

1. Requires only a working knowledge of Lebesgue integration. This is a real minimum since Fourier analysis cannot be dealt efficiently with less advanced integration theories.
2. Deals with both Fourier series and integrals (and the latter both in the one and multi-dimensional cases)
3. Starts from elementary topics and proceeds up to very advanced ones: just to get an idea, in the last chapter Banach Algebras and the Gel’fand transform are introduced, while distributions are introduced in the preceding one.

Having understood the theory as presented in this book opens the understanding of more advanced treatises (like the ones of Elias Stein and coauthors).

References

[1] Baggett, Larry; Fulks, Watson, Fourier analysis, Boulder, Colorado: API Anjou Press, Inc. pp. VIII, 183 (1979), ISBN: 0-88446-001-0, MR0554082, Zbl 0437.42001

I read mostly Springer books. You can try "Classical Fourier Analysis".

However, I do not think one book is ever sufficient to learn a subject thoroughly. Read also books about linear algebra as they explain concept about orthonormal basis, which is important for discreet Fourier transform. Do not hesitate to research from other sources including this website.

n.p. I always find it weird that most schools teach Fourier transform first and then its inverse (some of them do not teach the inverse at all). I learned inverse Fourier first and it made me understand better why we use Fourier transform.