Fourier Analysis and Its Applications

Thank God for this book. Folland is the grand-master of my life. I'm a grad student in applied math, studying for qualifying exams, and this book has been extremely helpful to me because it is perfectly rigorous and also crystal clear. When I say Folland is The One, I am referring to The Matrix. This book is that good. His treatment of distributions (otherwise known as "generalized functions") is especially nice.

I've been asked to teach a course on Fourier analysis,I knew nothing on the subject so I took about 15 books on this subject and went over all of them. My conclusion was that Folland's book is the best!. He explain the theory yet never forget for a minute the intuitive side of the subject. The book contains almost all the important issues and notions of the subject. If you have a solid background in vector calculus and you know some basic facts about ODE this is a very good book to learn the subject from. Moreover the book give the reader some of the important motivations to the basic ideas of functional analysis such as generating functions distrbutions it gives the connection also between linear algebra and the basic ideas that lies at the foundations for understanding normed function spaces and more. Moreover the book draw the line , in a very elegant way, between functional analysis PDE and Fourier analysis. Main subjects are:Fourier series,orthogonal sets Fourier and Laplace transforms,convolution, generating functions,Green functions, and more. Very recommended!.

Concepts are not hidden under obscure mathematical notation: they are stated explicitly in plain english and illustrated with examples. I read a couple of other books on this topic (and PDE) without really understanding the subtleties. With this book everything becomes magically clear and obvious -as you read- and don't feel like you need take another course in real analysis to understand this topic. Bonus: you get solutions to exercices.