Distribution Theory Books: Intro to Grad/Adv. Undergrad Math

[n!kofeyn]

I was wondering if anyone had any good suggestions on distribution theory books. I am looking for one that would be a good solid introduction for a graduate student in mathematics (or even an advanced undergraduate book). Example topics would include smooth test functions with compact support, distributions, examples of distributions, equivalent definitions of distributions, Schwartz structure theorem, Frechet space, semi-norms, support of a distribution, patching distributions, distributions with compact support, Schwartz functions, Fourier transforms, tempered distributions, etc.

I have found a few books, but decided not to list them to see what others like enough to suggest. I would use such a book to complement a course right now, and then hopefully later, to write some introductory notes of my own.

 

[George Jones]

As short introductions, I like chapters in two books by Folland:

chapter 9, Generalized Functions, from the undergrad text Fourier Analysis and Its Applications;

chapter 9, Elements of Distribution Theory, from the grad text Real Analysis: Modern Techniques and Their Applications.

Introduction to the Theory of Distributions by Friedlander and Joshi is the only book devoted to distributions that I have looked, but that was a long time ago (when Friedlander was the sole author, I think), and I remember very little about it. I think that it is meant for undergrads, and that it doesn't include all the topics that you listed.

 

[Landau]

Friedlander provides a very good introduction. Hormänder's Volume I is the classic reference (but hard to read). Basic+Advanced Real Analysis by Knapp also has some material on distributions, as does Rudin's Functional Analysis.

 

[jbunniii]

A good introduction is Strichartz, "A Guide to Distribution Theory and Fourier Transforms," which is a little bit loose with the rigor but is good for gaining an understanding of most of the topics you listed, without getting bogged down in technicalities. It doesn't even require Lebesgue theory, but it's a very useful book whether or not you know Lebesgue theory. I wish this book had been my first exposure to distributions.

I also liked what I read of Rudin's "Functional Analysis," although I've never owned it and have only read bits and pieces of library copies. If you like Rudin's style (I do) then this is well worth a look.

Hormänder, "The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis" is the only book I encountered that contained all the distribution material we covered in my graduate functional analysis course (and it contains much more, of course), so I'm partial to it as a reference. As Landau said, it's not light reading at all. But it is a very nice book and it's certainly not impossible to learn from if you have a good grounding in Lebesgue theory: it even has hints and answers to all of the exercises!

 

[n!kofeyn]

Thanks a lot for the suggestions everyone! Another book along the lines of Friedlander I found was Generalised Functions by Hoskins. It spends a good deal of time providing examples, properties, and behavior of the delta function, which is nice to see.

I also found Generalized Functions by Gelfand and Shilov, which looks to be good. There are multiple volumes in the series, though I've only seen the first volume. The library didn't have the others. The second volume is subtitled Function and Generalized Functions spaces, which might contain some of the topics I mentioned.

I haven't had the time to go through any of these books very seriously, as my course moved off of distributions quickly after I posted this, but I wanted to list them here for completeness.

Thanks again.