History of Approximation Theory (HAT)
Historical Papers
The following are seminal papers in approximation theory. Some of these
papers are given as pdf files. As such you will need an appropriate Adobe
Acrobat reader which you probably have. Otherwise go to Adobe. Others of these
papers are linked. If you have problems with the links, please let us
know.
- 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.
- 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.
- Chebyshev, P. L.,
Sur les questions de minima qui se rattachent
à la représentation approximative des fonctions,
Mém. Acad. Sci. Pétersb. 7 (1859), 199-291.
Also to be found in Oeuvres de P. L. Tchebychef, Volume 1, 273-378,
Chelsea, New York, 1961, from where this paper was scanned.
- 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 Jahrbuch Database
JFM.
- 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).
- Mittag-Leffler, G., Sur la
représentation analytique des fonctions d'une variable
réelle, Rend. Circ. Mat. Palermo 14 (1900),
217-224. Mittag-Leffler's proof of the Weierstrass Theorem. Also contains
a long, long footnote where there is given an explanation by
Phragmén of how the Weierstrass Theorem follows from work of
Runge. Review in Jahrbuch Database
JFM.
- 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 Jahrbuch Database
JFM.
- 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 Jahrbuch Database JFM.
- 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
Jahrbuch Database JFM.
- 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 Jahrbuch Database
JFM.
- 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 Jahrbuch Database
JFM.
- 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. Review in Jahrbuch Database
JFM.
You are invited and encouraged to send us your suggestions for
additional papers to post. However please note that we are limited by
copyright laws.
- Borel, É., Leçons sur les fonctions de variables
réelles et les
développements en séries de polynômes, Gauthier-Villars, Paris, 1905
[2nd edition appeared in 1928]. The first textbook devoted mainly to
approximation theory.
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