Topics in analysis and distribution theory (Q2761075)

This booklet has its origin in lectures the author gave to graduate students at the Ovidus University of Constanca/Romania in the years 1997 to 1999. It contains an interesting selection from advanced analysis. The six chapters are entitled ``Differentiable functions'', ``Approximation theorems'', ``The Fourier transformation'', ``Holomorphic functions and applications'', ``Distribution theory'' and ``Fundamental solutions''. Most of the chapters contain -- besides material the reader would expect under the heading -- also one or two highlights. These are: extension theorems, division theorems and Morse functions in chapter one, the Baouendi-Treves approximation theorem in chapter two, the Paley-Wiener(-Schwartz)-theorem in chapter three, quasianalytic functions and Phragmen-Lindelöf-theorems in chapter four, distributions as boundary values and wave front sets in chapter five. In chapter six the author first presents some direct methods to calculate special fundamental solutions and then two proofs of the Malgrange-Ehrenpreis theorem. Neither the profound and extensive work of \textit{N. Ortner} and \textit{P. Wagner} on fundamental solutions [e.g. Acta Appl. Math. 47, No. 1, 101-124 (1997; Zbl 0889.35020), and ``Functional Analysis'', S. Dierolf, S. Dineen, P. Domanski (eds.), Proceedings of the first international workshop helf at Trier University, Germany, September 26--October 1, 1994, 343-352 (1996; Zbl 0990.35036)], nor the quite ingenious work of \textit{H. König} [Proc. Am. Math. Soc. 120, No. 4, 1315-1318 (1994; Zbl 0792.35021)] are mentioned in this chapter. The author's approach to tempered fundamental solutions is via the Bernstein-Sato polynomial. This calls for some algebraic preparations. The presentation of the results in this booklet is adequate and thus makes it a useful basis for a seminar or a lecture on advanced analysis.