Polynomial Approximation. Just over 100 years ago K. Weierstrass [1885] proved that every continuous function on a closed interval is the uniform limit of polynomials; also, 75 years ago two basic works of S. N. Bernstein [19121,2] were published. The first con- tains his famous proof of Weierstrass's theorem, using ideas from probability theory. The second is a Prize Memoir , p u b l i s h e d by the Belgian A c a d e m y of Sciences, on quantitative questions in the uniform ap- proximation of continuous functions by polynomials. The same year saw the publ ica t ion of D u n h a m Jackson's fundamental work [1912] on approximation theory, containing the results of his dissertation [1911] written under Edmund Landau. Finally, 50 years ago we have M. H. Stone [1937] where, among other things, Weierstrass's theorem is generalized to the set- ting of continuous functions on a compact Hausdorff space (the Stone-Weierstrass theorem).

Our story begins earlier, January 28, 1853, with the p r e s e n t a t i o n by P. L. Ceby~ev 1 to the Imperial Academy of Sciences, St. Petersburg, of an article enti- tled "Th6orie des m6canismes connus sous le nom de parall61ogrammes" [1853]. He was motivated by me- chanics, in particular the work of James Watt on steam engines (the problem of converting circular motion into rectilinear motion). In an effort to design more ef- ficient machines, he was led to the problem of approx- imating a continuous function as closely as possible on a closed interval by polynomials of given degree. A big error at even one point was to be avoided, so the ap-

1 I am told that "~eby~ev" is p ronounced "~eby~ov," wi th the ac- cent on the last syllable. There are m a n y transliterations of this name: in English "~" becomes " s h , " whereas in French and German it becomes "ch" and " sch . " Likewise, "~" becomes "ch" in English, and " tch" " t sch" in French and German. Personally I favor the use of ~ and ~. I am told that wh i l e W. Feller was editor, Mathematical Reviews decided to use t hem for transliterating Cy~llic names. He was of Croatian origin, and Croatian is wri t ten with these symbols. This has n o w been changed; " s h " and "ch" are used, and all re- views are publ i shed in English.

Karl Weierstrass Edmund Landau

proximation was in the uniform norm. For differen- tiable functions one has the partial sums of the Taylor series, which are accurate near one point but not over an interval. He discusses modifications to obtain op- timal approximation over the interval, and is led to the problem of finding the best approximation to x n over [ -1 ,1 ] by polynomials of degree less than n. This leads to what are nowadays known as the Ceby~ev polynomials. These ideas are given a more detailed treatment in Ceby~ev [1859].

A famous result of Ceby~ev states that if p is a poly- nomial of degree at most n that is a best uniform ap- proximant to a continuous function f on a closed in- terval, then there are at least n + 2 points in the in- terval where ~ - Pl attains its maximum value. In the 1853 paper (w this result is stated as follows (here U denotes the polynomial): "the difference f - U has, as one knows, the following property: Among the largest and smallest values of the difference f - U between

* C o l u m n editor's address: Depar tment of Mathematics, University of Michigan, A n n Arbor, MI 48109 USA

THE MATHEMATICAL INTELLIGENCER VOL. 9, NO. 3 �9 1987 Springer-Veflag New York 5

Carl Runge Marshall Stone

the limits x = a - h, x = a + h, the same absolute value occurs at least n + 2 times." Ceby~ev goes on to use this result without proof; it is proved in the second paper (Th6or~me 2). In both papers he assumes that f is a differentiable function; this is used in the proof.

Ceby~ev never discusses the existence of a closest polynomial; he seems to regard this as evident (the uniqueness can be proved from the above theorem about n + 2 points). A modern reader would agree that the existence is obvious, because the role of com- pactness is now well understood, but this was not a lways the case. A dissertat ion by P. Kirchberger [1902] was devoted in part to reworking the theory with an eye to the new standards of rigor. 2 E. Borel [1905] bases his treatment on this dissertation: "La m6thode de Tchebicheff a 6t6 reprise et rendue rigou- reuse par M. Paul Kircherberger" (footnote 2, p. 82 in the 1928 edition, with the name misspelled).

Kirchberger [1903] seems to have been the first to use separa t ing hype rp l anes and the geomet ry of convex sets in the study of best approximation.

A modern reader, knowing Weierstrass's theorem, is surprised that Ceby~ev did not discover it. After all, Ceby~ev was concerned with much more detailed questions; however, he seems not to have asked what happens when the degree n tends to infinity.

Weierstrass considered a bounded continuous func- tion f on the real line. By convolving f with the func- tions {exp(- X2/E2)}/~.~ V2, ~ > O, he obtained entire func- tions that converge to f, uniformly on each finite in- terval, as e ~ 0. To obtain polynomials with this property, he took suitable partial sums of the power series. The theorem for functions defined on a finite interval follows from this. In the years that followed,

2 In the review of Kirchberger [1902] it is s tated tha t Poncelet first s u g g e s t e d the idea of s t u d y i n g bes t un i fo rm approximat ion; no ref- e rence is g iven, and I have f o u n d no o ther reference to this.

many other proofs were given; we have already men- tioned Bernstein's proof [19121]. Some details of earlier proofs are given in Chap. 4 of Borel [1905]. For ex- ample, one notes that every continuous function on [ -1 ,1] is the uniform limit of piecewise linear func- tions. The problem of approximating such functions by polynomials can be reduced to the problem of ap- proximating Ixl; that is, {X2} V2. Borel states that Le- besgue observed that this can be done by taking par- tial sums of the power series for {1 + y}'/~, y = x 2 - 1. (Borel does not give a reference for this observation of Lebesgue.)

C. Runge almost proved Weierstrass's theorem in- dependently in 1885 (I am indebted to Maurice Heins for pointing this out to me). On the one hand we have Runge's theorem [1884] on the uniform approximation of analytic functions on compact sets by rational func- tions. The proof involves representing the function by a Cauchy integral around a contour surrounding the compact set, then approximating the integral by Rie- mann sums (thereby obtaining rational functions), and finally moving the poles. He arranges the proof so that the rational functions are uniformly close to zero on another set. This allows him to construct, in an arbi- trary region G, functions that are analytic in G but not continuable into any larger domain (which was his motivation for writing the paper). Mittag-Leffier had obtained the latter result somewhat earlier by other methods, but he (as editor) published Runge's paper in Acta Mathematica anyway because the result was ob- tained independen t ly and because of the elegant method. (See the editor's footnote in [1884], p. 229.)

Runge [1885] points out that if f and g are contin- uous on [0,A], A-> 1, and if f(1) = g(1), t h e n g + (f - g)h n is uniformly close to f in [0,1] and to g in [1,A]. Here h,(x) = {1 + x2"} -1. I f f and g can each be uniformly approximated by rational functions, then one obtains uniform rational approximations to the combined function (that is, to f on the first interval and to g on the second). Developing this idea, he shows that a piecewise linear function on a closed interval is the uniform limit of rational functions. Then he points out that arbitrary continuous functions can be approx- imated by piecewise linear functions. He does not take the final step of replacing the rational functions by polynomials, although his method for moving poles [1884], p. 236, would have allowed him to do this.

Subsequently Jackson in his dissertation [1911] ob- tained a quantitative form of the Weierstrass theorem. Let f be continuous on [a,b] with modulus of conti- nuity 00f, and let E,(I) denote the distance (in the su- premum norm) to the polynomials of degree at most n. Then

6 THE MATHEMATICAL INTELLIGENCER VOL. 9, NO. 3, 1987

Bernstein [19122] obtained a partial converse: if E n c/n ~ for all n and some fixed c~, 0 < oL < 1, then f sat- isfies a Lipschitz condit ion of order o~.

Jackson s tud ied u n d e r E d m u n d L a n d a u in G6t- t ingen; in the preface to his 1930 book he described h o w his d isser ta t ion began. We quote the re levant part.

One day about twenty years ago I was admitted to the study of Professor Landau, seeking advice as to a subject for a thesis. After some preliminary inquiries as to my ex- perience and preferences, he handed me a long sheet of paper, and directed me to take notes as he enumerated some dozen or fifteen topics in various fields of analysis and number theory, with a few words of explanation of each. He told me to think about them for a few days, and to select one of them, or any other problem of my own choosing, with the single reservation that I should not prove Fermat's theorem, an injunction which I have ob- served faithfully. Guided partly by natural inclination, perhaps , and partly by recollection of a course on methods of approximation which I had taken with Pro- fessor B6cher a few years earlier, I committed myself to one of the topics which Landau had proposed, an investi- gation of the degree of approximation with which a given continuous function can be represented by a polynomial of given degree. When I reported my choice, he said meditatively, in words which I remember vividly in sub- stance, if not perfectly as to idiom: "Das ist ein schOnes Thema, ich beneide Sie um das T h e m a . . . Nein, ich be- neide Sie nicht, aber es ist ein wundersch6nes Thema!" [Beneiden means "to envy."]

Just 50 years ago Stone [1937], pp. 453-481, ex- t ended the Weierstrass theorem as follows. Let X be a compact Hausdor f f space and let (1 be a subalgebra of the real cont inuous functions on X. If et contains the constant funct ions and separates the points of X, then ct is un i fo rmly dense in C(X). Stone [1948] gives a su rvey of this and other generalizations; see also Ka- ku tan i [1941]. L. deBranges [1959] gives a beaut i fu l proof of the Stone-Weierstrass theorem; his me thod of p roo f has p r o v e d useful in fu r the r general iza t ions (see, for example, Gamelin [1969], Chap. 2, w

For a superb presenta t ion of the classical theory, see Na tanson [1949]; this book is t ruly a pleasure to read. For ~ e b y ~ e v ' s w o r k in mechan i c s see G e r o n i m u s [19541.

References

Abbreviations: Jrb., Zbl., and MR denote, respectively, Jahrbuch /iber die Fortschritte der Mathematik, Zentralblatt Kir Mathematik, and Mathematical Reviews.

S. N. Bernstein [19121], D~monstration du th6or~me de Weierstrass fond6e sur le calcul des probabilit6s, Khar'kov, Soobg~. matem, obg~. (2) 13, 1-2. Jrb. 43, 301. Russian translation: Collected works I, 105-106.

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- - [ 1 8 9 9 - 1 9 0 2 ] , Oeuvres, St. Petersburg. Reprinted: Chelsea Pub. Co., New York (1962). MR 26, 4870. Col- lected works in Russian translation: Poln. sobr. so~. Akad. Nauk SSSR, Vol. 2, 3 (1944-1951).

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S. Kakutani [1941], Concrete representation of abstract (M)- spaces. (A characterization of the space of continuous functions.), Ann. of.Math. 42, 994-1024. MR 3, 205.

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- - [ 1 9 0 3 ] , Ober Tschebychefsche Annhherungsmeth- oden, Math. Ann. 57, 509-540. Jrb. 34, 438-439.

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C. Runge [1884], Zur theorie der eindeutigen analytischen Functionen, Acta Math. 6, 229-244. Jrb. 17, 379-381.

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