Distributions: Theory and Applications

Name: Distributions: Theory and Applications
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J.J. Duistermaat · Johan A.C. Kolk

2010年8月 · Springer Science & Business Media

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I am sure that something must be found. There must exist a notion of generalized functions which are to functions what the real numbers are to the rationals (G. Peano, 1912) Not that much effort is needed, for it is such a smooth and simple theory (F. Tre`ves, 1975) In undergraduate physics a lecturer will be tempted to say on certain occasions: “Let ?. x/ be a function on the line that equals 0 away from 0 and is in?nite at 0 in such a way that its total integral is 1. The most important property of ?. x/ is exempli?ed Z by the identity 1 . x/?. x/ dx D . 0/; 1 where is any continuous function of x. ” Such a function ?. x/ is an object that one frequently would like to use, but of course there is no such function, because a function that is 0 everywhere except at one point has integral 0. All the same, it is important to realize what our lecturer is trying to accomplish: to describe an object in terms of the way it behaves when integrated against a function. It is for such purposes that the theory of distributions, or “generalized functions,” was created. It can be formulated in all dimensions, its mathematical scope is vast, and it has revolutionized modern analysis.

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Hans Duistermaat was a geometric analyst, who unexpectedly passed away in March 2010. His research encompassed many different areas in mathematics: ordinary differential equations, classical mechanics, discrete integrable systems, Fourier integral operators and their application to partial differential equations and spectral problems, singularities of mappings, harmonic analysis on semisimple Lie groups, symplectic differential geometry, and algebraic geometry. He was (co-)author of eleven books.

Duistermaat was affiliated with the Mathematical Institute of Utrecht University since 1974 as a Professor of Pure and Applied Mathematics. During the last five years he was honored with a special professorship at Utrecht University endowed by the Royal Netherlands Academy of Arts and Sciences. He was also a member of the Academy since 1982. He had 23 PhD students.

Johan Kolk has published in the areas of harmonic analysis on semisimple Lie groups, the theory of distributions, and classical analysis. Jointly with Duistermaat, he has written four books: besides the present one, one on Lie groups, and another on multidimensional real analysis. Until his retirement in 2009, he was affiliated with the Mathematical Institute of Utrecht University. For more information, see http://www.staff.science.uu.nl/~kolk0101/