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Weierstrass and uniform approximation. (Catalan. English summary) Zbl 1284.01031

From the summary: “Our aim is to present the Weierstrass theorem about uniform approximation of continuous functions by polynomials in the setting of the Weierstrass construction of the mathematical analysis, based on the representation of functions as sums of power series or of analytical functions, as well as his effort to introduce rigor with his program called arithmetization of analysis, that was on the basis of his criticism of Riemann’s methods. We also include the most interesting proofs of the approximation theorem that appeared after the Weierstrass work, many of them by some of his students or followers.”
The paper is subdivided into the following sections:
1. Karl Weierstrass (1815, Ostenfelde – 1897 Berlin); 2. Weierstrass/Riemann; 3. Convergència uniforme; 4. 1885: el teorema d’aproximació de Weierstrass; 4. 1891: Picard i l’aproximació a partir de l’equació de Poisson; 6. 1898: Lebesgue i l’aproximació per poligonals; 7. 1901: el fenomen de Runge; 8. 1902: Fejér i l’aproximació per mitjanes de Fourier; 9. 1908: Landau presenta una demonstració senzilla; 10: 1911: el mètode probabilistic de Bernstein; 11. Els estudiants de Weierstrass; References.
Besides the historical and mathematical flow as presented above, there are short curricula vitae of Riemann, Picard, Lebesgue, Runge, Fejér, Landau, Bernstein, with photographs, together with a condensed survey of their works in respect to that of Weierstrass. In Section 11, a list of 42 doctoral students is presented and, in addition, photographs of Mittag-Leffler, Du Bois-Reymond, Kovalevskaya, Weierstrass, together with stories about works of the latter form.
The paper is written in Catalan. Anyway, the reviewer found the paper pretty reading.

MSC:

01A55 History of mathematics in the 19th century
01A70 Biographies, obituaries, personalia, bibliographies
31-03 History of potential theory
32-03 History of several complex variables and analytic spaces
40A10 Convergence and divergence of integrals
40A30 Convergence and divergence of series and sequences of functions
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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