# Further studies on Fourier Series and Integrals.

If you had to choose two books from the following list, which pair would you chose, and why? If you haven't read any, would you pick any pair among the list based on the author of the book? I am trying to pick two, but I need some guidance! The list is from the end of Apostol's Mathematical Analysis chapter on Fourier Theory, titled "References for Further Studies". This is what Apostol has covered:

• Best Approximation, Fourier series relative to an orthonormal system. Properties of the Fourier coefficients (Bessel, Parseval). Riesz Fischer.
• Convergence and representation problems for trigonometric series. Riemann Lebesgue lemma. Dirichlet Integrals.
• Integreal Representation for the partial sums. Riemann's localization theorem.
• Sufficient condition for pointwise convergence (Dini, Jordan). Cesaro summability. Fejér's theorem. Consequences. Weiertrass' Approximation theorem.
• Other forms of Fourier series (exponential form, general periods)
• The Fourier integral theorem. Exponential form of the Fourier integral theorem. Integral transforms. Convolutions.
• The Poisson summation formula.

1. Carslaw, H. S., Introduction to the Theory of Fourier's Series and Integrals, $3^{\rm rd}$ ed. Macmillan, London 1930.
2. Edwards, R. E., Fourier Series, A Modern Introduction, Vol 1. Holt, Rinehart and Winston, New York, 1967.
3. Hardy, G. H., and Rogosinski, W. W., Fourier Series. Cambridge Universiy Press, 1950.
4. Hobson, E. W., The Theory of Functions of a Real Variable and the Theory of Fourier's Series, Vol. 1, $3^{\rm rd}$ ed. Cambridge Universiy Press, 1927.
5. Indritz, J. Methods in Analysis. Macmillan, New York, 1963.
6. Jackson, D., Fourier Series and Orthogonal Polynomials. Carus Monograph NO. 6. Open Court, New York, 1941.
7. Rogosinski, W. E., Fourier Series. Chelsea, New York, 1941.
8. Titchmarsh, E. C., Theory of Fourier Integrals. Oxford University Press, 1937.
9. Wiener, N., The Fourier Integral. Cambridge Universiy Press, 1933.
10. Zygmund, A., Trigonometrical Series, $2^{\rm nd}$ ed. Cambridge Universiy Press, 1968.

[I am suspecting this should be made community Wiki]

• I don't know any of those books, but for me the standard reference would be Katznelson. – Alex Youcis Aug 11 '13 at 3:15
• @AlexYoucis What is the name of the book? – Pedro Tamaroff Aug 11 '13 at 3:15
• amazon.com/… – Alex Youcis Aug 11 '13 at 3:20
• A lot of the listed books seem to contain introductions to Fourier analysis while the title says "further studies", so it is not clear what you are aimed for. If you've studied Fourier series once already, then Katznelson is the main text people go to next. Perhaps get a library or pdf copy and try push through it a little more. I haven't read anything on that list either, and I'm not sure in the value of picking based on author but if I were to, Hardy and Titchmarsh certainly stand out. – Ragib Zaman Aug 11 '13 at 3:29
• Similarly to Alex, I've also never read any of those books. But I would second his suggestion for Katznelson's book. I would add Stein and Shakarch, Fourier Analysis. I hate choosing based off of a name (isn't there a saying along those lines?), but from the list I'd pick Zygmund's book. I've glanced at a few times while referencing chasing, but it came off rather dry. – dls Aug 11 '13 at 3:52

@Peter, I strongly recommend T.W. Körner's Fourier Analysis (see http://www.amazon.com/Fourier-Analysis-T-246-rner/dp/0521389917/ref=sr_1_1?s=books&ie=UTF8&qid=1376232360&sr=1-1&keywords=korner+fourier+analysis) and the accompanying book of exercises. It's exceedingly well-written, full of interesting stuff, and accessible to good undergraduates.

• It looks really amazing! Thanks! – Pedro Tamaroff Aug 11 '13 at 17:16

(Too long for a comment) I hadn't known you learned from Apostol before. From what I recall, Apostol allows you into certain topics without heavy prerequisites by slightly restricting generality in some cases. For example my guess is that you didn't need to know measure theory, and he only proves a version of Riemann-Lebesgue integrating over a finite interval $[a,b]?$

My main advice to you is to put Fourier series down for a little bit of time, let it stew in your brain while you learn some other important areas of analysis because they are essential to further study. Measure theory and Functional analysis (pre reqs to that are metric spaces and general topology) are crucial not just for Fourier series but most topics in analysis.

After you know those topics, you can skip all those listed books and work through Katznelson. I really think that is the best plan. Sure you can delay learning measure theory and functional analysis and learn a little more about fourier series without them, but eventually you will need them and if you have them, the same material in the more elementary books are more easily proved and stored in one's head.

• He proves Riemann Lebesgue for any interval –– open, closed, bounded, unbounded –– but your're quite right: all is done without any measure theory or whatever. – Pedro Tamaroff Aug 11 '13 at 4:22

I strongly recommend Champeney, A Handbook of Fourier Theorems even though he omits the proofs. It is up to date, very complete, and reliable. If you want to learn the proof of one of those results, he gives a reliable reference. All the definitions are explained.