Sergei Natanovich Bernstein (1880-1968)

  • Alternative spellings: Bernshtein, Bernshteyn
  • Bernstein, S. N., Démonstration du Théorème de Weierstrass fondée sur le calcul des Probabilités, Comm. Soc. Math. Kharkov 2.Series XIII No.1 (1912), 1-2. This is Bernstein's famous paper where he presented a probabilistic proof of the Weierstrass Theorem, and introduced what we today call Bernstein polynomials. Note that his proof is somewhat "overinvolved". We nowadays present this proof in a slightly more elegant form. This paper is reprinted in Russian in Bernstein's collected works. Note that the bound volume XIII of the journal carries the year 1913 even though the first few numbers published separately each carry the year 1912.
  • Bernstein, S. N., Sur l'ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné, Mem. Cl. Sci. Acad. Roy. Belg. 4 (1912), 1-103. This paper was awarded a prize by the Belgian Academy of Science. This was as a consequence of his answer to a question posed by de La Vallée Poussin. Bernstein proved that it is not possible to approximate |x| in [-1,1] by a polynomial of degree n with an approximation of order greater than 1/n. It also contains the first form of what we call inverse theorems, Bernstein's inequality and more.
  • Bernstein, S. N., O nailuchshem priblizhenii nepreryvnykh funktsii posredstvom mnogochlenov dannoi stepeni Comm. Soc. Math. Kharkov 2.Series, XIII No. 2-5, (1912), 49-194, and here is the somewhat changed version of the above paper, as it appears in Bernstein Collected Works, Constructive Function Theory 1905-1930, Akademia Nauk SSSR, 1952, 11-104. Note that the bound volume XIII of the journal carries the year 1913 even though the first few numbers published separately each carry the year 1912. Here is Bernstein's public lecture given during the defense of his doctoral dissertation in Kharkiv (modern spelling) on May 19, 1913.
  • Akhiezer, N. I., Academician S. N. Bernstein and his Work on the Constructive Theory of Functions (in Russian), Izdat. Harkov. Gosudarstv. Univ., Kharkov, 1955; translated into German as Das Akademiemitglied S. N. Bernstein und seine Arbeiten zur konstruktiven Funktionentheorie, translated and with a preface by Ralitza K. Kovacheva and Heinz H. Gonska. in Mitt. Math. Sem. Giessen No. 240 (2000), xiv + 97 pages. The original is an expanded form of the survey paper by N. I. Akhiezer with the same title which appeared in Uspehi Mat. Nauk (N.S.) 6 (1951), no. 1(41), 3-67, which was written by Akhiezer on the occasion of Bernstein's 70th birthday.
  • Aleksandrov, P. S., N. I. Akhiezer, B. V. Gnedenko, and A. N. Kolmogorov, Sergei Natanovich Bernstein: Obituary, (in Russian) Uspekhi Mat. Nauk 24 (3) (1969), 211-218, translation in Russia Math. Surveys 24 (3) (1969), 169-176.
  • Garloff, J., A. P. Smith, Preface, in Special Issue on the Use of Bernstein Polynomials in Reliable Computing: A Centennial Anniversary of Reliable Computing 17 (2012), i-vi.
  • Stark, E. L., Bernstein-Polynome 1912-1955, Functional Analysis and Approximation (Oberwolfach 1980), Internat. Ser. Numer. Math. 60, Birkhäuser, Basel-Boston, 1981, pp.443-461. Discusses Bernstein's introduction of Bernstein polynomials in his 2-page paper on Weierstrass' Theorem and the resulting literature, including the problem of dating Bernstein's paper.
  • Videnskii, V. S., Sergei Natanovich Bernshtein - Founder of the constructive theory of functions. Uspehi Mat. Nauk 16 1961 no. 2 (98), 21-24. In English in Russian Math. Surveys 16 1961, 17-20. See here.

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