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Weierstrass and approximation theory. (English) Zbl 0968.41001
This is a lovely article on Weierstrass and the early development of approximation theory. It begins with a short biography of Weierstrass. Two main themes stand out in his work: To set a new standard of rigor in analysis, and his love for power series or more generally for function series. The first theme is documented by his construction of a continuous, nowhere differentiable function which was shocking to the mathematical community at the time. Weierstrass presented this in his lectures since 1861 but published his example (using a cosine series) in 1872. Further history on that by Bolzano, Riemann, Takagi, and du Bois-Reymond is mentioned. The second theme is documented by the Fundamental Theorem of Approximation Theory: Algebraic polynomials are dense in $C[a,b]$, where $-\infty <a<\sigma <\infty$. This was published by Weierstrass. in 1885 when he was 70 years old, and proved by representing $f\in C[a,b]$ as a limit of integrals $\int^\infty_{-\infty}$ depending on a parameter $k$. Thus $f$ is the uniform limit of a sequence of entire functions and hence of a sequence of polynomials. Weierstrass. states and proves also the analogous theorem about the density of trigonometric polynomials. The author then lists and analyses further proofs (before 1913) of the Fundamental Theorem. He puts them into three groups. In Group 1 there are proofs based on singular integrals (Weierstrass, Picard, Fejér, Landau), while those in Group 2 are based on the approximation of a particular function, like a polygonal function (Runge, Lebesgue, Mittag-Leffler, Lerch). Left over are those in Group 3 by Bernstein, Volterra, Lerch. It is interesting to note that Runge proved (also in 1885!) that rational functions are dense in $C[a,b]$ but overlooked the fact that this is true already for polynomials. Lebesgue reduces the Fundamental Theorem to the special case $f(x)= |x|$, and he raises (1908) apparently for the first time questions about the speed of approximation, three years before Jackson’s dissertation appeared. The last section deals with various generalizations: Müntz’s theorem, Hermite-Fejér interpolation, Carleman’s theorem, Stone-Weierstrass, and Bohman-Korovkin. All these theorems are given with full explanation, proofs, as well as historical notes. It is clear that this article is necessary reading for all approximators.

MSC:
41-02Research monographs (approximations and expansions)
WorldCat.org
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References:
[1] Achieser, N. I.: Theory of approximation. (1956) · Zbl 0072.28403
[2] Baillaud, B.; Bourget, H.: Correspondance d’hermite et de Stieltjes. (1905)
[3] Banach, S.: Über die baire’sche kategorie gewisser funktionenmengen. Stud. math. 3, 174-179 (1931) · Zbl 57.0305.05
[4] Bell, E. T.: Men of mathematics. (1936) · Zbl 0014.33902
[5] Bernstein, S. N.: Sur LES recherches récentes relatives à la meilleure approximation des fonctions continues par LES polynomes. Proc. of 5th inter. Math. congress 1, 256-266 (1912)
[6] Bernstein, S. N.: Sur l’ordre de la meilleure approximation des fonctions continues par LES polynomes de degré donné. Mém. cl. Sci. acad. Roy. belg. 4, 1-103 (1912)
[7] Bernstein, S. N.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Comm. soc. Math. kharkow 13, 1-2 (1912/13)
[8] Biermann, K. R.: Karl Weierstrass. Dictionary of scientific biography (1976)
[9] Bohman, H.: On approximation of continuous and of analytic functions. Ark. mat. 2, 43-56 (1952) · Zbl 0048.29901
[10] Du Bois-Reymond, P.: Versuch einer classification der willkürlichen functionen reeller argumente nach ihren aenderungen in den kleinsten intervallen. J. reine angew. Math. 79, 21-37 (1875)
[11] Du Bois-Reymond, P.: Untersuchungen über die convergenz und divergenz der fourierschen darstellungsformeln. Abh. math.-phys. Classe K. Bayerische akad. Wissenschaften 12, 1-13 (1876)
[12] Bolzano, B.: Paradoxes of the infinite. (1950) · Zbl 0039.00506
[13] Borel, É.: Lecons sur LES fonctions de variables réelles et LES développements en séries de polynomes. (1905)
[14] Borwein, P.; Erdélyi, T.: Polynomials and polynomial inequalities. (1995) · Zbl 0840.26002
[15] Bourbaki, N.: Topologie générale (Livre III). Espaces fonctionnels dictionnaire (Chapitre X). (1949) · Zbl 0036.38601
[16] Boyer, C. B.; Merzbach, U. C.: A history of mathematics. (1989)
[17] Brosowski, B.; Deutsch, F.: An elementary proof of the stone--Weierstrass theorem. Proc. amer. Math. soc. 81, 89-92 (1981) · Zbl 0482.46014
[18] Buck, R. C.: Studies in modern analysis. (1962) · Zbl 0128.24104
[19] Burckel, R. B.; Saeki, S.: An elementary proof of the müntz--szász theorem. Expo. math. 4, 335-341 (1983) · Zbl 0525.41036
[20] Butzer, P. L.; Nessel, R. J.: Aspects of de la vallée poussin’s work in approximation and its influence. Arch. hist. Exact sci. 46, 67-95 (1993) · Zbl 0782.01013
[21] Butzer, P. L.; Stark, E. L.: 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”. Edmund Landau, collected works, 83-111 (1986)
[22] Butzer, P. L.; Stark, E. L.: ”Riemann’s example” of a continuous nondifferentiable function in the light of two letters (1865) of Christoffel to Prym. Bull. soc. Math. belgique 38, 45-73 (1986) · Zbl 0629.01013
[23] Cakon, R.: Alternative proofs of Weierstrass theorem of approximation: an expository paper. (1987)
[24] Carleman, T.: Sur un théorème de Weierstrass. Ark. mat., ast. Fysik B 20, 1-5 (1927) · Zbl 53.0237.02
[25] Cheney, E. W.: Introduction to approximation theory. (1966) · Zbl 0161.25202
[26] Dieudonné, J.: Foundations of modern analysis. (1969) · Zbl 0176.00502
[27] Faber, G.: Über die interpolatorische darstellung stetiger funktionen. Jahresber. deut. Math. verein 23, 190-210 (1914) · Zbl 45.0381.04
[28] Feinerman, R. P.; Newman, D. J.: Polynomial approximation. (1974) · Zbl 0309.41006
[29] Fejér, L.: Sur LES fonctions bornées et intégrables. CR heb. Séances acad. Sci. Paris 131, 984-987 (1900)
[30] Fejér, L.: Ueber interpolation. Nachr. gesell. Wiss. göttingen math. Phys. kl., 66-91 (1916) · Zbl 46.0419.01
[31] Fejér, L.: Über weierstrasssche approximation, besonders durch hermitesche interpolation. Math. ann. 102, 707-725 (1930) · Zbl 56.0255.02
[32] Frih, E. M.; Gauthier, P. M.: Approximation of a function and its derivatives by entire functions of several variables. Canad. math. Bull. 31, 495-499 (1988) · Zbl 0641.32011
[33] Gaier, D.: Lectures on complex approximation. (1987) · Zbl 0612.30003
[34] Gerver, J.: The differentiability of the Riemann function at certain rational multiples of ${\pi}$. Amer. J. Math. 92, 33-55 (1970) · Zbl 0203.05904
[35] Gerver, J.: More on the differentiability of the Riemann function. Amer. J. Math. 93, 33-41 (1970) · Zbl 0203.05904
[36] Grabiner, J. V.: The origins of Cauchy’s rigorous calculus. (1981) · Zbl 0517.01002
[37] Gray, J. D.: The shaping of the Riesz representation theorem: A chapter in the history of analysis. Arch. hist. Exact sci. 31, 127-187 (1984) · Zbl 0549.01010
[38] Hardy, G. H.: Weierstrass’s non-differentiable function. Trans. amer. Math. soc. 17, 301-325 (1916) · Zbl 46.0401.03
[39] Hildebrandt, T. H.: A simple continuous function with a finite derivative at no point. Amer. math. Monthly 40, 547-548 (1933) · Zbl 59.0285.03
[40] Hille, E.: Analytic function theory. (1962) · Zbl 0102.29401
[41] Hunt, B. R.: The Hausdorff dimension of graphs of Weierstrass functions. Trans. amer. Math. soc. 126, 791-800 (1998) · Zbl 0897.28004
[42] Jackson, D.: Über die genauigkeit der annäherung stetiger funktionen durch ganz rationale funktionen gegebenen grades und trigonometrische summen gegebener ordnung. (1911) · Zbl 42.0434.03
[43] Jackson, D.: The general theory of approximation by polynomials and trigonometric sums. Bull. amer. Math. soc. 27, 415-431 (1921) · Zbl 48.0289.01
[44] Jackson, D.: The theory of approximation. Amer. math. Soc., colloquium publ. 11 (1930)
[45] Jackson, D.: A proof of Weierstrass’s theorem. Amer. math. Monthly 41, 309-312 (1934) · Zbl 60.0211.01
[46] Kaplan, W.: Approximation by entire functions. Michigan math. J. 3, 43-52 (1955/56)
[47] Korovkin, P. P.: On convergence of linear positive operators in the space of continuous functions. Dokl. akad. Nauk SSSR 90, 961-964 (1953)
[48] Korovkin, P. P.: Linear operators and approximation theory. (1960) · Zbl 0094.10201
[49] Kowalewski, G.: Über bolzanos nichtdifferenzierbare stetige funktion. Acta math. 44, 315-319 (1923) · Zbl 49.0174.01
[50] Kuhn, H.: Ein elementarer beweis des weierstrassschen approximationssatzes. Arch. math. 15, 316-317 (1964) · Zbl 0127.29103
[51] Kuratowski, C.: Topologie. (1958) · Zbl 0078.14603
[52] Landau, E.: Über die approximation einer stetigen funktion durch eine ganze rationale funktion. Rend. circ. Mat. Palermo 25, 337-345 (1908) · Zbl 39.0472.02
[53] Lebesgue, H.: Sur l’approximation des fonctions. Bull. sci. Math. 22, 278 (1898)
[54] Lebesgue, H.: Sur la représentation approchée des fonctions. Rend. circ. Mat. Palermo 26, 325-328 (1908) · Zbl 39.0473.01
[55] Lebesgue, H.: Sur LES intégrales singulières. Ann. fac. Sci. univ. Toulouse 1, 25-117 (1909) · Zbl 41.0327.02
[56] Lerch, M.: About the Main theorem of the theory of generating functions (in Czech). Rozpravy ceske akad. 33, 681-685 (1892)
[57] Lerch, M.: Sur un point de la théorie des fonctions génératrices d’abel. Acta math. 27, 339-351 (1903)
[58] Levasseur, K. M.: A probabilistic proof of the Weierstrass approximation theorem. Amer. math. Monthly 91, 249-250 (1984) · Zbl 0564.41005
[59] Luxemburg, W. A. J.; Korevaar, J.: Entire functions and müntz--szász type approximation. Trans. amer. Math. soc. 157, 23-37 (1971) · Zbl 0224.30049
[60] MacTutor, Available at, http://www-groups.dcs.st-and.ac.uk/history.
[61] Mazurkiewicz, S.: Sur LES fonctions non dérivables. Stud. math. 1, 92-94 (1929)
[62] Méray, C.: Nouveaux exemples d’interpolations illusoires. Bull. sci. Math. 20, 266-270 (1986)
[63] Meyer, Y.: Wavelets: algorithms and applications. (1993) · Zbl 0821.42018
[64] Mittag-Leffler, G.: Sur la représentation analytique des functions d’une variable réelle. Rend. circ. Mat. Palermo 14, 217-224 (1900) · Zbl 31.0409.01
[65] Müntz, C. H.: Über den approximationssatz von Weierstrass. (1914) · Zbl 45.0633.02
[66] Nachbin, L.: Elements of approximation theory. (1976) · Zbl 0331.46015
[67] Narasimhan, R.: Analysis on real and complex manifolds. (1968) · Zbl 0188.25803
[68] Natanson, I. P.: Constructive function theory. (1964) · Zbl 0133.31101
[69] Neuenschwander, E.: Riemann’s example of a continuous ’nondifferentiable’ function. Math. intelligencer 1, 40-44 (1978) · Zbl 0374.26002
[70] Ostrowski, A.: Vorlesungen über differential-und integralrechnung. (1951) · Zbl 0044.27501
[71] Oxtoby, J. C.: Measure and category. Gtm 2 (1986) · Zbl 0217.09201
[72] Picard, E.: Sur la représentation approchée des fonctions. C. R. Heb. séances acad. Sci. Paris 112, 183-186 (1891)
[73] Picard, E.: Traité d’analyse. (1891)
[74] Picard, E.: Lectures on mathematics. Clark university 1880--1899 decennial celebration, 207-259 (1899)
[75] Prolla, J. B.: Weierstrass--stone, the theorem. (1993) · Zbl 0849.41029
[76] Ransford, T. J.: A short elementary proof of the Bishop--stone--Weierstrass theorem. Math. proc. Cambridge philos. Soc. 96, 309-311 (1984) · Zbl 0537.41018
[77] De Rham, G.: Sur un exemple de fonction continue sans dérivée. Enseign. math. 3, 71-72 (1957) · Zbl 0077.06104
[78] Rivlin, T. J.: The Chebyshev polynomial. (1974) · Zbl 0299.41005
[79] Rogers, L. C. G.: A simple proof of müntz’s theorem. Math. proc. Cambridge philos. Soc. 90, 1-3 (1981) · Zbl 0469.41007
[80] Rudin, W.: Real and complex analysis. (1966) · Zbl 0142.01701
[81] Runge, C.: Zur theorie der eindeutigen analytischen functionen. Acta math. 6, 229-244 (1885)
[82] Runge, C.: Über die darstellung willkürlicher functionen. Acta math. 7, 387-392 (1885/86)
[83] Runge, C.: Über empirische funktionen und die interpolation zwischen äquidistanten ordinaten. Zeit. math. Physik 46, 224-243 (1901) · Zbl 32.0272.02
[84] Schwarz, H. A.: Zur integration der partiellen differentialgleichung \partial2u/\partialx2+\partial2u/\partialy2=0. J. reine angew. Math. 74, 218-253 (1871)
[85] Skrasek, J.: Le centenaire de la naissance de matyas lerch. Czech math. J. 10, 631-635 (1960)
[86] Spivak, M.: Calculus. (1994) · Zbl 0159.34302
[87] Stark, E. L.: Bernstein--polynome, 1912--1955. Isnm 60, 443-461 (1981)
[88] Stone, M. H.: Applications of the theory of Boolean rings to general topology. Trans. amer. Math. soc. 41, 375-481 (1937) · Zbl 0017.13502
[89] Stone, M. H.: A generalized Weierstrass approximation theorem. Math. magazine 21, 167-184 (1948)
[90] Stone, M. H.: A generalized Weierstrass approximation theorem. Studies in modern analysis, 30-87 (1962)
[91] Szabados, J.; Vértesi, P.: Interpolation of functions. (1990) · Zbl 0721.41003
[92] Szász, O.: Über die approximation stetiger funktionen durch lineare aggregate von potenzen. Math. ann. 77, 482-496 (1916) · Zbl 46.0419.03
[93] Sz.-Nagy, B.: Introduction to real functions and orthogonal expansions. (1965) · Zbl 0128.05101
[94] Takagi, T.: A simple example of a continuous function without derivative. Proc. physico-math. Soc. Japan 1, 176-177 (1903) · Zbl 34.0410.05
[95] Timan, A. F.: Theory of approximation of functions of a real variable. (1963) · Zbl 0117.29001
[96] Todd, J.: Introduction to the constructive theory of functions. Caltech lecture notes (1961) · Zbl 0096.32304
[97] Ullrich, P.: Anmerkungen zum ”riemannschen beispiel” \sum\inftyn=1sinn2xn2 einer stetigen, nicht differenzierbaren funktion. Result. math. 31, 245-265 (1997)
[98] De La Vallée Poussin, Ch.J.: Sur l’approximation des fonctions d’une variable réelle et leurs dérivées par des polynomes et des suites limitées de Fourier. Bull. acad. Royale belgique 3, 193-254 (1908) · Zbl 39.0329.02
[99] De La Vallée Poussin, Ch.J.: Sur la meilleure approximation des fonctions d’une variable réelle par des expressions d’ordre donné. CR acad. Sci. Paris 166, 799-802 (1918) · Zbl 46.0416.02
[100] De La Vallée Poussin, Ch.J.: L’approximation des fonctions d’une variable réelle. L’enseign. math. 20, 5-29 (1918) · Zbl 46.0416.03
[101] De La Vallée Poussin, Ch.J.: Leçons sur l’approximation des fonctions d’une variable réelle. (1919) · Zbl 47.0908.02
[102] Volterra, V.: Sul principio di Dirichlet. Rend. circ. Mat. Palermo 11, 83-86 (1897) · Zbl 28.0363.01
[103] Van Der Waerden, B. L.: Ein einfaches beispiel einer nicht-differenzierbaren stetigen funktion. Math. Z. 32, 474-475 (1930) · Zbl 56.0929.02
[104] Walsh, J. L.: Interpolation and approximation by rational functions in the complex domain. (1935) · Zbl 0013.05903
[105] K. Weierstrass, Über continuierliche Functionen eines reellen Arguments, die für keinen Werth des letzteren einen bestimmten Differentialquotienten besitzen, Königliche Akademie der Wissenschaften, 18 Juli 1872. [Also in ”Mathematische Werke,” Vol. 2, pp. 71--74, Mayer & Müller, Berlin, 1895.]
[106] K. Weierstrass, Zur Functionenlehre, Monatsber. Königl. Akad. Wiss, 1880. [Also in ”Mathematische Werke,” Vol. 2, pp. 210--223, Mayer & Müller, Berlin, 1895.]
[107] K. Weierstrass, Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen, Sitzungsber. Akad. Berlin, 1885, pp. 633--639, 789--805. [This appeared in two parts. An expanded version of this paper with ten additional pages also appeared in ”Mathematische Werke,” Vol. 3, 1--37, Mayer & Müller, Berlin, 1903.]
[108] Weierstrass, K.: Sur la possibilité d’une représentation analytique des fonctions dites arbitraires d’une variable réelle. J. math. Pure appl. 2, 105-113 (1886)
[109] Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. amer. Math. soc. 36, 63-89 (1934) · Zbl 0008.24902
[110] Yamaguti, M.; Hata, M.; Kigami, J.: Mathematics of fractals. AMS transl. Math. monographs (1997) · Zbl 0888.58030