History of Approximation Theory (HAT)
Articles on the History of Approximation Theory
The following are papers concerning various topics in the history of
approximation theory. Some of these papers are linked. If you have
problems with the links, please let us know. Some of these papers are
given as pdf files. For them you will need an appropriate Adobe Acrobat
reader which you probably have. Otherwise go to Adobe.
- 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 Uspekhi Mat. Nauk (N.S.)
6 (1951), no. 1(41), 3-67, which was written by Akhieser on the occasion
of Bernstein's 70th birthday.
- Akhiezer, N. I., List of
Publications. A list of publications of Naum Il'ich Akhiezer.
We don't know how exact this list is.
- 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 Russian
Math. Surveys 24 (3) (1969), 169-176.
- Anderson, J. M., Obituary.
The short obituary to J. Milne Anderson that appeared in Constructive
Approximation 43 (2) (2016).
- Andreev, A., V. I. Berdyshev, B. Bojanov, B. S Kashin,
S. V. Konyagin, S. M. Nikolskii, K. I. Oskolkov,
P. Petrushev, Bl. Sendov, S. A. Telyakovskii, and V. N. Temlyakov,
In memory of Sergei Borisovich Stechkin [1920-1995],
East J. Approx. 2 (1996), 131-133.
- Aptekarev, A. I., G. López Lagomasino, E. B. Saff, V. Totik,
and H. Stahl, Andrei Aleksandrovich
Gonchar (on his 80th
birthday), Uspekhi Mat. Nauk 66 (2011), 197-204. English
transl. in Russian Math. Surveys 66 (2011), 1209-1216, see
here.
- Askey, R. A., and P. Nevai,
Gabor Szegö: 1895--1985 , Math.
Intelligencer 18, 3 (1996), 10-22, (for those with access) and
in Hungarian, Mat.
Lapok 3 (1993), 13-32.
- Babich, V. M., A. M. Vershik, S. V. Videnskii,
O. L. Vinogradov, I. K. Daugavet, N. Yu. Dodonov, V. V. Zhuk, B. M.
Makarov, A. N. Podkorutov, Yu. G. Reshetnyak, M. A. Skopina,
V. L. Fainshmidt, V. P. Havin, and N. A. Shirokov,
Garal'd Isidorovich Natanson. (Russian) Uspekhi Mat. Nauk
59 (2004), no. 4 (358),
181-185; translation in
Russian Math. Surveys 59 (2004), no. 4, 771-776,
in English (for those with access).
- Basharin, G. P., A. N.Langville and
V. A. Naumov,
The life and work of A.A. Markov,
Linear Algebra and its Applications 386
(2004) 3--26. This article should be read with some caution, according to
its review
in Zentralblatt Math. For those with access, this article may be viewed at
Science Direct.
- Boas, R. P., Extremal problems for polynomials, Amer. Math.
Monthly 85 (1978), 473-475. A short pleasant expository note
about the origins of the Markov inequality. This article may be viewed
at JSTOR
- Bogoljubov, A. N., A. T. Grigorʹjan, G. I. Drinfelʹd,
and A. P. Juškevič, Ja. L. Geronimus (on the occasion of his
seventy-fifth birthday), (Russian) Voprosy Istor. Estestvoznan. i
Tehn. Vyp. 2(43) (1973), 71-72.
- Bolibrukh, A. A., A. G. Vitushkin, V. S. Vladimirov, E. F.
Mischenko, S. P. Novikov, Yu. S. Osipov, A. G. Sergeev, P. L. Ul’yanov, L.
D. Fadeev, and E. M. Chirka, Andrei Aleksandrovich Gonchar (on his 70th
birthday), Uspekhi Mat. Nauk 57 (2001), 185–190. English
transl. in Russian Math. Surveys 57 (2001), 191–198, see
here.
- de Boor, C., and A. Pinkus, The B-spline recurrence relations of
Chakalov and Popoviciu, J. Approx. Theory 124 (2003),
115-123. This article may be viewed at
Science Direct.
- Borwein, P., and T. Erdélyi, Polynomials and Polynomial
Inequalities, Springer-Verlag, New York, 1995. Contains a great deal
of information on the history of approximation theory scattered throughout
the text.
- Braess, D. Helmut Werner, Computing 36 (1986),
181-182. For those with access, this article can be found
here.
- Braess, D., and R. Schaback, Helmut
Werner, Jahresbericht DMV 89 (1987), 179-195. This
article can be found in this issue
of the Jahresbericht.(The article starts on page 39 of this pdf file.)
- Brezinski, C., Historical Perspective on Interpolation,
Approximation and Quadrature, in Handbook of Numerical Analysis,
Volume III, eds. P. G. Ciarlet and J. L. Lions, p. 3-46, North Holland,
1994.
- Brudnyi, Y., An excerpt
concerning Timan's life from: Timan's type result on approximation by
algebraic polynomials, in Operator Theory, Advances and
Applications, Vol.98, Birkhäuser Verlag, Basel, 1997,
pp.92-101.
- Bruins, E. M., On the history of approximation, in Constructive
Function Theory '77, Sofia, 1980, 275-280.
- Burkill, J. C., Charles-Joseph de la Vallée Poussin, J.
London Math. Soc. 39 (1964), 165-175.
- Butzer, P. L., An outline of the life and work of E. B.
Christoffel (1829-1900), Historia Math. 8 (1981), 243-276.
For those with access, this article may be viewed at
Science Direct.
- Butzer, P. L., A retrospective on 60 years of approximation theory
and associated fields, J. Approx. Theory, 160 (2009), 3-18.
For those with access, this article may be viewed at
Science Direct.
- Butzer, P., and F. Jongmans, P. L. Chebyshev (1821-1894), a guide
to his life and work, J. Approx. Theory 96 (1999), 111-138.
For those with access, this article may be viewed at
Science Direct.
- Butzer, P. L., and F. Jongmans, P. L. Chebyshev (1821-1894) and
his contacts with Western European scientists, Historia Math.
16 (1989), 46-68. For those with access, this article may be viewd
at
Science Direct.
- Butzer, P. L., and R. J. Nessel, Aspects of de La Vallée
Poussin's work in approximation and its influence, Archive Hist. Exact
Sciences 46 (1993), 67-95. For those with access, this article
may be viewed at
SpringerLink.
- Butzer, P. L., G. Schmeisser, and R. L. Stens, James Gourlay Clunie, 1926-2013.
Excerpted from their paper: The classical and approximate sampling
theorems and their equivalence for entire functions of exponential type,
J. Approx. Theory 179 (2014), 94-111.
- Butzer, P. L., G. Schmeisser, and R. L. Stens, Qazi Ibadur Rahman, 1934-2013.
Excerpted from their paper: The classical and approximate sampling
theorems and their equivalence for entire functions of exponential type,
J. Approx. Theory 179 (2014), 94-111.
- Butzer, P. L., M. Schmidt, and E. L. Stark, Observations on the
history of central B-splines, Archive Hist. Exact Sciences
39
(1988), 137-156. For those with access, this
article may be viewed at
SpringerLink
- Butzer, P. L., and E. L. Stark, The singular integral of Landau
alias the Landau polynomials - Placement and impact of Landau's article
``Über die Approximation einer stetigen Funktion durch eine ganze
rationale Funktion'', in Edmund Landau, Collected Works, Volume 3,
eds. P. T. Bateman, L. Mirsky, H. L. Montgomery, W. Schall, I. J.
Schoenberg, W. Schwarz, and H. Wefelscheid, 83-111, Thales-Verlag,
Essen.
- Butzer, P. L., and R. L. Stens, Sampling theory for not
necessarily band-limited functions: a historical overview, SIAM
Review 34 (1992), 40-53. For those with access, this article may
be viewed
electronically at
SIAM Review and at JSTOR.
- Cheney, E. W., Introduction to Approximation Theory,
McGraw-Hill Book Co., New York, 1966. The notes at the end of this book,
pages 224-233, are especially recommended.
- Cuyt, A., A bibliography of the works of Prof. Dr. H. Werner,
J. Comp. and Applied Math. 19 (1987), 3-8. For those with
access, this article can be viewed at
Science Direct.
- Daepp, U, Gauthier, P., Gorkin, P., and G. Schmieder, Alice in
Switzerland: The Life and Mathematics of Alice Roth, The Mathematical
Intelligencer 27 (2005), 41-53. For those with access, this
article may be viewed at
Springerlink.
- Deutsch, F., The convexity of Chebyshev sets in Hilbert space, in
Topics in Polynomials of One and Several Variables and Their
Applications, eds. Th. M. Rassias, H. M. Srivastava, and A.
Yanushauskas, World Scientific Publ. Co., 1993, 143-150. A brief
historical account of work on the open question of whether every
Chebyshev set in a Hilbert space must be convex.
- Deutsch, F., Best Approximation in Inner Product Spaces,
CMS Book in Mathematics, Springer-Verlag, New York, 2001. Each chapter
ends with an interesting and useful section of ``Historical Notes''.
- Elliott, D., Lagrange interpolation -- decline and fall?,
Internat. J. Math. Edu. Sci. Tech. 10 (1979), 1-12.
- Erdelyi, T., and P. Vértesi,
In Memoriam: Paul Erdős (1913-1996),
J. Approx. Theory 94 (1998), 1-41. The article also contains
a bibliography of Erdős' work in approximation theory compiled by J.
Szabados. For those with access, this article may be viewed at
Science Direct.
- Farouki, R. T.,
The Bernstein polynomial basis: a centennial retrospective,
Comput. Aided Geom. Design 29 (2012), no. 6, 379–419.
- 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.
- Gasca, M., and T. Sauer, On the history of multivariate polynomial
interpolation. Numerical analysis 2000, Vol. II: Interpolation and
extrapolation, J. Comput. Appl. Math. 122 (2000), 23-35.
For those with access, this article may also be viewed at
Science Direct
- Gautschi, W., A survey of Gauss-Christoffel quadrature formulae,
in E. B. Christoffel, The Influence of his Work on Mathematics and the
Physical Sciences, eds. P. L. Butzer, F. Feher, 72-147,
Birkhäuser, Basel, 1981.
- Golinskii, L., On the Scientific Legacy of Ya. L. Geronimus,
Self-Similar Systems (Dubna, 1998), 273-281, Joint Inst.
Nuclear Res., Dubna, 1999. An English translation by the author is
here.
- Goncharov, V. L.,
The Theory of Best Approximation of Functions,
J. Approx. Theory 106 (2000), 2-57.
This article, here translated into English, was originally published in
1945 in the collection The Scientific Heritage of P. L. Chebyshev. I.
Mathematics, 122-172, Nauka. See also the commentary by V. M.
Tikhomirov on pages 58-65 which follows the above article. For those with
access, this article may be viewed at
Science Direct.
The commentary by V. M. Tikhomirov may be viewed at
Science Direct.
- Hsu, L. C., A survey of some recent developments of approximation
theory in China, in Approximation Theory, IV (College Station,
Tex., 1983), eds. C. K. Chui, L. L. Schumaker and J. D. Ward,
123-151, Academic Press, New York, 1983.
- Kotlyar, B. D., S. M. Nikolskii, V. G. Ponomarenko,
V. M. Tikhomirov, V. N. Trofimov, and P. L. Ulyanov, Aleksandr
Filippovich Timan, (Russian) Uspekhi Mat. Nauk 44 (1989),
no. 3(267), 145-146; translation in Russian Math. Surveys
44 (1989) no. 3, 179-180. The English translation of this article
can be viewed at IOPScience.
- López Lagomasino, G., A. F. Martinez, P. Nevai, and E. B.
Saff,
In Memoriam: Andrei Aleksandrovich Gonchar:
November 21, 1931 – October 10, 2012,
J. Approx. Theory 172 (2013), A1-A13. For those with
access, this article may be viewed at
Science Direct.
- Lorentz, G. G., Superpositions, metric entropy, complexity of
functions, widths, Bull. London Math. Soc. 22 (1990), 64-71.
This is one of the articles within a series dedicated to the
work of Andrei Nikolaevich Kolmogorov which appeared in Bull. London
Math. Soc. 22 (1990), 31-100. See here for
the full article.
- Lorentz, G. G., Mathematics and Politics in the Soviet Union
from 1928 to 1953, J. Approx. Theory 116 (2002), 169-223.
For those with access, this article may be viewed at
Science Direct.
- Lyubich, Yu. I., Naum Il'ich
Akhiezer, (in Russian) Teor. Funktsii Funktsional. Anal. i
Prilozhen. 56 (1991), no. 4, 3-14; translation in J. Math.
Sci. 76 (1995), no. 4, 2441–2451.
- Marchenko, V. A., Yu. A. Mitropol'skii, V. A. Pogorelov, A.
M. Samoilenko, I. V. Skrypnik, E. Ya. Khruslov, Naum
Il'ich Akhiezer (on the centenary of his birth), (in Russian)
Ukrain. Mat. Zh. 53 (2001), no. 3, 291-293; translation in
Ukrainian Math. J. 53 (2001), no. 3, 333–335.
- Micchelli, C. A., On the work of Theodore J. Rivlin, in "The
heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday
of T. J. Rivlin", Ann. Numer. Math. 4(1-4) (1997)
viii-xiv.
- Mitrpol'skii, Yu. A., V. K. Dzyadyk, and V. T. Gavrilyuk,
Evgen'ii Yakovlevich Remez (on the occasion of the ninetieth
anniversary of his birth), (in
Russian) Ukrain. Mat. Zh. 38 (1986), 539-540.
- Nevai, P., Geza Freud, Orthogonal Polynomials and Christoffel
Functions (A Case Study), J. Approx. Theory 48 (1986),
3-167. For those with access, this article may be viewed at
Science Direct.
- Nikol'skii, S. M., A.A. Privalov and P.L. Ul'yanov, Pavel
Petrovich Korovkin (on his seventieth birthday), Uspekhi Mat. Nauk
39:1 (1984), 179-180; translation in Russian Maths. Surveys
39:1 (1984), 209-211. This article in Russian can be viewed here, and in
English (for those with access)
here.
- Nikol'skii, S. M., A.A. Privalov and P.L. Ul'yanov, Pavel
Petrovich Korovkin (Obituary), Uspekhi Mat. Nauk
41:5 (1986), 155-156; translation in Russian Maths. Surveys
41:5 (1986), 119-121. This article in Russian can be viewed
here, and in English (for those with access)
here.
- Schaback, R., Helmut Werner, a vita written
specifically
for this website.
- Schoenberg, I. J., List of
Publications, essentially as it appears in I. J. Schoenberg,
Selected Papers, ed. C. de Boor, Volume I, pp. xi-xviii,
Birkhäuser, Boston, 1988.
- Schoenberg, I. J., A brief account of
my life and work, in I. J. Schoenberg, Selected Papers, ed. C.
de Boor, Volume I, 1-10, Birkhäuser, Boston, 1988.
- Sewell, W. E., Remarks on the mathematical researches of J. L.
Walsh, SIAM J. Numer. Anal. 3 (1966), 344-348. For those
with access, this article
may be viewed at
SIAM J. Numer. Anal.
and at JSTOR.
- Siegmund-Schultze, R., Der Beweis des Weierstraßschen
Approximationssatzes 1885 vor dem Hintergrund der Entwicklung der
Fourieranalysis (German) [The proof of Weierstrass' approximation theorem
of 1885 against the background of the development of Fourier analysis],
Historia Math. 15 (1988), 299-310. For those with access,
this article may be viewed at
Science Direct.
- 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.
- Steffens, K.-G., The History of Approximation Theory From
Euler
to Bernstein, Birkhäuser, Boston, 2006.
- Stone, M., A reminiscence on the extension of the Weierstrass
approximation theorem, Historia Math. 3 (1976), 328. For
those with access,
this article may be viewed at
Science Direct.
- Szegö, G., The contribution of L. Fejér to the
constructive
function theory, in Proceedings of the Conference on Constructive
Theory of Functions, eds. G. Alexits and S. B. Stechkin,
19-26, Budapest, 1972.
- Tikhomirov, V. M., A. N. Kolmogorov and approximation theory, (in
Russian) Uspekhi Mat. Nauk 44 (1989), 83-122; translation in
Russian Math. Surveys 44 (1989), 101-152.
- Tikhomirov, V. M., The papers of P. L. Chebyshev on approximation
theory, (Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh.
119 (1995), 86-90; translation in Moscow Univ. Math. Bull.
50 (1996), 39-42.
- Todd, J., Applications of transformation theory: a legacy from
Zolotarev (1847-1878), in Approximation theory and Spline
Functions, eds. S. P. Singh, J. H. W. Burry, and B. Watson,
207-245, Reidel, Dordrecht, 1984.
- Todd, J., A legacy from E. I. Zolotarev (1847-1878), Math.
Intelligencer 10 (1988), 50-53.
- Turán, P., On some open problems of approximation theory,
J. Approx. Theory 29 (1980), 23-85. For those with access,
this article may be viewed at
Science Direct. Pages 86-89, (for those with access
here) just following this paper, contain a list of articles dealing
with Turán's Problems as presented in the above 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.
-
Vinogradov, O. L., V. V. Zhuk, V. L. Fainshmidt, and V. P. Khavin,
Garal'd Isidorovich Natanson, Proceedings of the St. Petersburg
Mathematical Society, X (2005), 237–252, also appears in
Amer. Math. Soc. Transl. Ser. 2, 214, Amer.
Math. Soc., Providence, RI, 2005.
- Vlasov, V. F., A. N. Kolmogorov, V. G. Ponomarenko, V. N.
Trofimov, and B. D. Judin, Aleksandr Filippovic Timan (on the occasion
of his
sixtieth birthday). (Russian) Uspekhi Mat. Nauk 36 (1981),
no. 2(218), 221-226.
- Vulih, B. Z., M. K. Gavurin, and
S. M. Lozinskii,
Isidor Pavlovic Natanson. Obituary. (in Russian)
Uspehi Mat. Nauk 20 (1965), no. 1 (121), 171-175. In
English (for those with access).
- Werner, I., In Memoriam, pages VII-IX of Rational Approximation
and its Applications in Mathematics and Physics, (Lancut, 1985),
331–350, LNM 1237, Springer,
Berlin, 1987.
- Widder, D. V., Joseph Leonard Walsh, SIAM J. Numer. Anal.
3 (1966), 171-172. For those with access, this article may be viewed
electronically at
SIAM J. Numer. Anal. and at
JSTOR.
- Wynn, P., The work of E. B. Christoffel on the theory of
continued fractions, in E. B. Christoffel, The Influence of his Work
on Mathematics and the Physical Sciences, eds. P. L. Butzer, F.
Feher, 190-202, Birkhäuser, Basel, 1981.
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