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Collected works. Vol. 3. Ed. by Leopold Schmetterer and Karl Sigmund. (English, German) Zbl 0881.01046

Wien: Springer-Verlag. xiii, 581 p. (1997).
[For the reviews of Vols. 1 and 2, see Zbl 0859.01030 and Zbl 0847.01033.]
Contents of this last volume of the collection: Reprints of papers on measure theory and integration (commented by H. Bauer); on singular integrals and Fourier analysis (J.-P. Kahane); one single paper on functions of two complex variables (L. Kaup); philosophical writings (Ch. Thiel); and several testimonials of biographical interest.
Written from 1903 to 1933 Hahn’s work in analysis, on the eve of more abstract approaches, stands on the zenith of the Weierstrassian real analysis. His admiration for this comes out in his long eulogy, reprinted here among the philosophical writings, on Pringsheim’s soon forgotten lectures. Hahn’s work in general harmonic analysis, based on pointwise convergence, was soon superseded by that of Wiener, using \(L^2\)-convergence. On more than hundred pages on singular integrals Hahn meticulously elaborates properties of delta sequences, as they had been used in early work of Cauchy and Poisson, unknown to Hahn. Even after distribution theory these representations of delta functions deserve to be better known. Some of Hahn’s work on measure theory, as his last paper on products of additive set functions (1933), was of lasting influence.
Though many of Hahn’s philosophical papers were already reprinted elsewhere the careful philological comments by Ch. Thiel justify the collection here. Hahn is remembered as the founder, center and teacher of the Vienna Circle and for his lucidly written expositions of the analytical approach to philosophical problems. A short paper on Occam’s razor presents his philosophy in a nutshell (pp. 457-478, headline missing here).
The very last lines of the collection, written by Hahn’s former student Olga Taussky-Todd, run: “…his overall contribution to mathematics will remain outstanding among those of the Austrian mathematicians of his time”.

MSC:

01A75 Collected or selected works; reprintings or translations of classics
01A60 History of mathematics in the 20th century
28-03 History of measure and integration
42-03 History of harmonic analysis on Euclidean spaces
00A30 Philosophy of mathematics
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