Historical Papers

The following are seminal papers in approximation theory.

  • 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.
  • Chebyshev, P. L., Théorie des mécanismes connus sous le nom de parallélogrammes, Mém. Acad. Sci. Pétersb. 7 (1854), 539-568. Also to be found in Oeuvres de P. L. Tchebychef, Volume 1, 111-143, Chelsea, New York, 1961, from where this paper was scanned. The surprisingly many and varied linkages designed by Chebyshev can all be viewed, in action, at Mechanisms (pointing your cursor at the Russian flag at the upper right corner of that page gives you the opportunity to choose to see the page in English). Each of the many mechanisms shown can be activated by a click; there are plans to provide the detailed comments associated with the animation eventually in English.
  • Fejér, L., Sur les fonctions bornées et intégrables, Comptes Rendus Hebdomadaries, Seances de l'Academie de Sciences, Paris 131 (1900), 984-987 (in equation (2), 1/2 cos --> 1/2 + cos). This fundamental paper formed the basis of Fejér's doctoral thesis obtained in 1902 from the University of Budapest, under the supervision of H. A. Schwarz. Fejér was 20 years old when this paper appeared. The paper contains the "classic" theorem on Cesaro (C,1) summability of Fourier series, thus providing a direct constructive proof of Weierstrass' Theorem.
  • Fejér, L., Über Interpolation, Nachrichten der Gesellschaft der Wissenschaften zu Göttingen Mathematisch-physikalische Klasse, 1916, 66-91. This is where Fejér introduced what we now call the Hermite-Fejér Interpolation operator (based on the zeros of the Chebyshev polynomial), and proved the uniform convergence of the sequence of these polynomials to the function being interpolated.
  • Kakeya, S., On approximate polynomials, Tohoku Math. J. 6 (1914), 182-186. This paper is about uniform approximation by polynomials with integer coefficients. Kakeya continues the work of Pál by finding necessary and sufficient conditions on the functions f defined on [-1,1] which can be so approximated. He also shows that on an interval of length at least 4, the only functions which can be approximated in this way are the integral polynomials themselves.
  • Kirchberger, P., Über Tchebychefsche Annäherungsmethoden, Dissertation. Univ. Göttingen, 1902. See also Math. Ann. 58 (1903), 509-540. This latter article does not contain the full results of his thesis. It discusses multivariate approximation problems. Kirchberger's "proof" of the alternation theorem, a theorem about the stability of the linear approximation operator and Chebyshev's approximation algorithm, i.e., all his one-dimensional results, are only to be found in the thesis. (Paul Kirchberger's doctoral advisor was David Hilbert.)
  • Lebesgue, H., Sur l'approximation des fonctions, Bull. Sciences Math.22 (1898), 278-287. Here is Lebesgue's beautiful proof of the Weierstrass Theorem. It is based on the idea of approximating the single function |x| by polynomials, and the fact that one can uniformly approximate any continuous function on a closed finite interval by continuous piecewise linear approximants. This is Lebesgue's first paper. He obtained his doctorate 4 years later. (This journal was also called the Darboux Bulletin.) Review in ZBMath Open.
  • Markov, A. A., Ob odnom voproce D. I. Mendeleeva, Zapiski Imperatorskoi Akademii Nauk SP6. 62 (1890), 1-24. This is the original paper in Old Russian spelling (and we thank V. V. Arestov and Elena Berdysheva for providing this copy). This paper contains the proof of the Markov inequality for algebraic polynomials. This journal was later called Mémoires de l'Academie Impériale des Sciences de St.-Pétersbourg VIe séries. Due, it seems, to translating the name of the journal into French and then back into Russian, many sources now reference this article as being in Izv. Petersburg Acad. Nauk or some variant thereof. As such it was referenced (and also with the year 1889) in A. A. Markov's Selected Works from 1948 which contains a transcription of the above paper into modern Russian spelling. We also have an English translation On a question by D. I. Mendeleev prepared by Carl de Boor and Olga Holtz.
  • Markov, V. A., O funktsiyakh, naimeneye uklonyayushchikhsya ot nulya v dannom promezhutke [On functions which deviate least from zero in a given interval], 1892. This was a preprint/treatise from the Department of Applied Mathematics, Imperial St.-Petersburg University. It was translated into German, with a short foreword by Bernstein, and appeared as: Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen, Math. Ann. 77 (1916), 213-258. The paper contains the proof of the Markov inequality for higher derivatives of algebraic polynomials. The appendix of V. A. Gusev in the book of E. V. Voronovskaja titled "The Functional Method and its Applications", Vol. 28 of Translations of Mathematical Monographs of the AMS, 1970, reproduces the final (and most essential) part of Markov's proof almost identically (even the letters in formulae are the same). Vladimir Andreyevich Markov was a younger half-brother of Andrey Andreyevich Markov. This paper was published while he was a 21 year old student at the St.-Petersburg University. He died at the age of 25 (of tuberculosis).
  • Müntz, Ch. H., Über den Approximationssatz von Weierstrass, in H. A. Schwarz's Festschrift, Berlin, 1914, pp. 303-312. This paper contains the original proof of the Müntz Completeness Theorem. It affirmatively answers a question posed by S. N. Bernstein two years previously. Review in ZBMath Open.
  • Newton, Sir Isaac, page 695, page 696 of `Philosophiae naturalis principia mathematica' (the original alongside an English translation), containing Lemma V of Book III in which Newton introduces divided differences in the construction of a polynomial interpolant to arbitrary data.
  • Pál, J., Zwei kleine Bemerkungen, Tohoku Math. J. 6 (1914), 42-43. The first paper to consider uniform approximation by polynomials with integer coefficients. Pál proves that if f is continuous on [-a,a], |a|<1, and f(0) is an integer then f may be uniformly approximated thereon by polynomials with integer coefficients.
  • Picard, E., Sur la représentation approchée des fonctions, Comptes Rendus Hebdomadaries, Seances de l'Academie de Sciences, Paris 112 (1891), 183-186. This is the first alternative proof of the Weierstrass Theorem. It also contains the first proof of the Weierstrass Theorem for functions of several variables. Review in ZBMath Open.
  • Riesz, F., Sur certains systèmes d'équations fonctionelles et l'approximation des fonctions continues, Comptes Rendus Acad. Sci. Paris 150 (1910), 647-677. Also appears in Oeuvres of F. Riesz on p. 403-406. There is a mix-up in some copies of the Oeuvres and there this paper is on pages 403, 404, 398 and 399. This is the first paper where it is stated and proved that an element of C([a,b]) is in the closure of a subspace if and only if every continuous linear functional that vanishes on the subspace also vanishes on the element.
  • Riesz, F., Über lineare Funktionalgleichungen, Acta Math. 41 (1918), 71-98. Contains first general proof of existence of best approximation from finite-dimensional subspace (see Hilfssatz 3, p. 77, i.e., Proposition 3 in the English translation of the relevant part of the paper).
  • Runge, C., Über die Darstellung willkürlicher Functionen, Acta Math. 7 (1885/86), 387-392. This paper contains a proof of the fact that any continuous function on a finite interval can be uniformly approximated by rational functions. Phragmén (see footnote in Mittag-Leffler's 1900 paper) pointed out that Runge's previous article (Acta Math. 6) contains a method of replacing the rational functions by polynomials. These papers do not explicitly contain Weierstrass' Theorem. Review in ZBMath Open.
  • Volterra, V., Sul principio di Dirichlet, Rend. Circ. Mat. Palermo 11 (1897), 83-86. Another proof of Weierstrass' Theorem for trigonometric polynomials. It is to be found right at the end of the paper. Review in ZBMath Open.
  • Weierstrass, K., Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsberichte der Akademie zu Berlin 633-639 and 789-805, 1885. Weierstrass' paper with his proof of the Weierstrass Theorem on density of algebraic polynomials in the space of continuous real-valued functions on any finite closed interval. Also the analogous result for trigonometric polynomials. An expanded version of this paper with ten additional pages appeared in Weierstrass' "Mathematische Werke", Vol. 3, 1-37, Mayer and Müller, Berlin, 1903. Review in ZBMath Open.
  • Weierstrass, K., Sur la possibilité d'une représentation analytique des fonctions dites arbitraires d'une variable réelle, J. Math. Pure et Appl. 2 (1886), 105-113 and 115-138. This is the translation of the Weierstrass 1885 paper and, as the original, it appeared in two parts and in subsequent issues, but under the same title. This journal was, at the time, called Jordan Journal.

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