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Distribution Theory: Convolution, Fourier Transform, and Laplace Transform

Walter de Gruyter
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Professor Gerrit van Dijk, Emeritus at the University of Leiden has, with De Gruyter’s assistance, produced an elegant slim volume on Distribution Theory (the text proper fits within a hundred pages), based on lectures given by the author to advanced undergraduates and beginning graduate students at Utretcht and Leiden. The paper, quality of printing and binding are of a good standard (much unlike a few recent publications by Springer-Verlag that the reviewer has perused) and the book sells for just over $30 on Amazon.

Each chapter begins with a short “Summary,” lists between one and three “Learning Targets,” and ends with a guide to “Further Reading.” Five of the nine main chapters include a small selection of exercises. The inclusion of learning targets is unusual in a textbook written at this level.

Many proofs are sketched and the reader is often referred to Schwartz’s Théorie des distributions (which in spite of its fame seems to lack an English translation) for more technical results. This style makes for pleasurable reading, especially for the mathematically mature reader. In fact, Van Dijk’s lectures may be used as a prelude or an opening act to a serious reading of (or course based on) Schwartz’s famous text.

However, the advanced undergraduate may not appreciate “For terminology and notations we generally follow Bourbaki.” As for

The main prerequisites for the book are elementary real, complex and functional analysis and Lebesgue integration. In the later chapters we assume familiarity with some advanced measure theory and functional analysis, in particular with the Banach-Steinhaus theorem.

most advanced undergraduates or beginning graduate students with such background (perhaps appearing with less frequency in the US than in Europe) may be inclined to tackle a more in-depth presentation than van Dijk’s. That said, the appendix summarizing this material is adequate and the text may serve such a student with a useful “Lonely Planet/Rough Guide” to distribution theory.

The lectures are elegantly written and guide the reader through a rapid tour of distribution theory, perhaps not unlike a quick mediterranean cruise through a carefully chosen succession of ports. It would probably take the advanced undergraduate in possession of the prerequisites about seven days (over a suitable break with less distractions than during term) to read the entire text in detail.

A few minor quibbles. The author comments in the preface:

There is relatively little expository literature on distribution theory compared to other topics in mathematics, but there is a standard reference [L. Schwartz, Théorie des distributions], and also [Gel’fand and Shilov, Generalized functions, Vol. 1]. I have mainly drawn on…

There are actually at least two introductions that the reviewer was familiar with (and probably many more beyond):

  1. Strichartz, Robert S. A guide to distribution theory and Fourier transforms. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1994. (See the MathSciNet review.)
  2. Duistermaat, J. J. and Kolk, J. A. C. Distributions: Theory and Applications. Translated from the Dutch by J. P. van Braam Houckgeest. Cornerstones. Birkhäuser Boston, Inc., Boston, MA, 2010. (See also the MathSciNet review.)

Both Strichartz’s and Duistermaat-Kolk’s books are aimed at a similar audience to van Dijk’s (though Duistermaat-Kolk is far more elementary in that it does not require the Lebesgue theory), but they are substantially longer and much better motivated than van Dijk’s lectures. (Wikipedia tells us that Duistermaat, apart from being born in Utrecht, returned and remained there from 1974 to his death in 2010. Little surprise that van Dijk was not aware of Duistermaat-Kolk’s text!) Before seeing van Dijk’s book if I were to recommend a text for an independent study, I would choose either Strichartz or Duistermaat-Kolk depending on the student’s inclinations. Now, van Dijk’s text gives me a reasonable outline for an (\(2 \leq n \leq 5\))-week ending to a Topics in Analysis course.

We end this review with another quote, this one from the final chapter, on Summable Distributions, which ends with the following Further Reading:

An application of this chapter is a mathematically correct definition of the Feynman path integral, see [E. G. F. Thomas, Path distributions on sequence spaces, preprint, 2000.].

The reviewer felt the urge to investigate further, and will refer the reader to Tony Dorlas’ piece in the following memorial to Thomas: “A good definition is half the work”, NAW 5/13, nr. 4, December 2012.

Tushar Das is an Assistant Professor of Mathematics at the University of Wisconsin – La Crosse.

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Wednesday, May 15, 2013
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Gerrit van Dijk
De Gruyter Graduate Lectures
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Wednesday, May 15, 2013