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I'm looking for a good book on distribution theory (in the Schwartz sense), I have the basic knowledge as given in Grafakos' Classical Fourier Analysis, but I want to know more about it. Is the reference still Laurent Schwartz' Théorie des distributions? That book is hard to obtain, it seems to be only available in France.

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    $\begingroup$ Some answers can be found at the corresponding MathOverflow question: mathoverflow.net/questions/20314/… $\endgroup$ – Jonas Meyer Dec 9 '10 at 19:55
  • $\begingroup$ mathoverflow.net/questions/20314/… $\endgroup$ – Robin Chapman Dec 9 '10 at 19:55
  • $\begingroup$ It isn't specifically about distributions, but there's a fair bit on that subject in Rudin's Functional Analysis. $\endgroup$ – Rotwang Dec 9 '10 at 19:56
  • $\begingroup$ @Rotwang: I have that book, it does not contain much more than Grafakos. $\endgroup$ – Jonas Teuwen Dec 9 '10 at 20:10
  • $\begingroup$ @Jonas: @Robin: Okay, to me it seems that Laurent Schwartz book wouldn't be a bad choice after reading that. The other suggestions for treatises are quite old as well, so I can as well take the book from the master. Now, I only have to find a source where I can purchase it... $\endgroup$ – Jonas Teuwen Dec 9 '10 at 20:12
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The book

Duistermaat J., Kolk J. Distributions: Theory and Applications (Birkhäuser 2010)

deserves to be mentioned.

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  • $\begingroup$ I agree. I found this book most useful as opposed to Strichartz' book which asks the reader to prove that so-and-so is a distribution without even formally defining what a distribution is. Actually, I would only recommend the latter if you are already familiar with distributions and just want something to read in bed. $\endgroup$ – Jason Born Apr 29 '17 at 15:55
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Theory of Distributions by J. Ian Richards and Heekyung K. Youn is a self-described "non-technical introduction", which seems to mean you don't need to know functional analysis, measure theory, or topology. But you do need to think more like a mathematician than like a physicist or engineer; it's all mathematically rigorous. It contains the authors' original results on the question of when two distributions can be multiplied.

Distribution Theory and Transform Analysis by A. H. Zemanian develops the theory, then does Fourier and Laplace transforms, then applies it all to problems arising in engineering.

And there's Introduction to Fourier analysis and generalised functions by Sir James Lighthill.

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Since I want to close this question, I will post an answer myself.

A Guide to Distribution Theory and Fourier Transforms - Strichartz is a nice introduction but it contains almost nothing.

Théorie des distributions by Laurent Schwartz is written by the master and father on the subject and therefore I say it is a good book.

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  • $\begingroup$ Have you looked at Friedlander and Joshi's introduction to the theory of distributions. It is a nice small book, with most of the topological vector space stuff distilled out. (If you want the TVS stuff, there are a few more other books in the MO thread mentioned above, e.g. Horvath's book...) $\endgroup$ – Willie Wong Dec 12 '10 at 1:10
  • $\begingroup$ @Willie Wong: I would like the TVS stuff. I'll look at the Horvath-book. $\endgroup$ – Jonas Teuwen Dec 13 '10 at 22:45
  • $\begingroup$ @JonasTeuwen Perhaps you could be able to answer this question (and maybe even earn the bounty, if you do so in the next two days): Reference request: Choquet theory. (Sorry for a comment which is unrelated to this question, but I wanted to ping you somewhere and I have not seen you in chat for some time.) $\endgroup$ – Martin Sleziak Oct 27 '16 at 1:05
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A book I've been reading that seems pretty good and is not listed at the link is Griffel's Applied Functional Analysis. Cheap too!

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I'm surprised no one has yet mentioned I. M. Gel'fand et al.'s five volume work Generalized Functions. Unfortunately, it is out of print (Volume I is going for 3000$ on Amazon at the time of writing), but if you can get your hands on a copy, I've found them to be an invaluable reference.

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I think the following books are useful and very interesting.

A Guide to Distribution Theory by Strichartz, Introduction to the Theory of Distributions, by Friedlander, Distributions Theory and Applications, by Duistermaat and Kolk.

Is this for your own interest or for your thesis? Or both? Just curious.

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  • $\begingroup$ Laurent Schwartz has another excellent book on this subject that is translated into English: " mathematics for the physical science", Dover publication. It is entirely about distribution theory and its applications. Very readable. $\endgroup$ – M. Rahmat Aug 20 '16 at 16:52

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