By Peter Duren
ISBN-10: 0821801309
ISBN-13: 9780821801307
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Example text
L. J. Nessel (2004), 381. See e. g. J. Mawhin (1992), 120. 8) 28 J. b a/ and then let ! 9). a/ in a way reminiscent of Riemann’s integration, except that Riemann’s integral corresponds, as easily verified, to the special choice of a constant mapping ı. This observation led Jaroslav Kurzweil68 and Ralph Henstock69 to define independently, some fifty years ago, a new type of integration. Namely, a function g W Œa; b ! 9). It can be shown that this integral is more general than Lebesgue’s and indeed equivalent to Denjoy–Perron’s integral.
Tamen, hoc non porta (ut, in modo vero erroneo, aliquo crede) ad relinque studios mathematico, que, ad contrario, recipe novo impulso. In vero, circa anno 1913, illo initia et cultiva cum grande successu novo campo de investigationes mathematico: illo dedicato ad Calculo numerico, neglecto et considerato quasi cum dedignatione ab aliquo mathematico moderno (per quanto jam mathematicos illustre [. . ] ne habe omisso de dedica scriptos ad quaestiones de calculo numerico)” (In 1903 Peano begins some philological studies and interlinguistic works, whom he devotes the greatest part of his activity till his death.
Almost all of them concern quadrature formulas which are exact for polynomials of a certain degree. The third is the most important, because the content goes beyond the title, discussing a general property of an important class of linear functionals. Here we shall be able to consider only this topic in detail for lack of space. It must be remarked that interesting results on quadrature formulas are already contained in the books 16: • Applicazioni geometriche del calcolo infinitesimale (1887); • Lezioni di Analisi infinitesimale (1893).
A Century of Mathematics in America by Peter Duren
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