WEIERSTRASS AND UNIFORM APPROXIMATION 1. Karl

WEIERSTRASS AND UNIFORM APPROXIMATION
JOAN CERDÀ
Abstract. We present the theorem of uniform approximation of continuous functions by polynomials in the Weierstrass framework for the construction of analysis
based on the representation of functions by sums of power series and analytic functions, and his effort to bring new standards of rigor to analysis that accompanied
his differences with Riemann. We also describe the most original proofs of the
approximation theorem that appeared after the Weierstrass death.
1. Karl Weierstrass (1815 Ostenfelde-1897 Berlin)
Known as the father of modern analysis,
Weierstrass devised tests for the convergence
of series and contributed to the theory of periodic functions, functions of real variables,
elliptic functions, Abelian functions, converging infinite products, and the calculus of
variations. He also advanced the theory of
bilinear and quadratic forms.
Encyclopaedia Britannica
While his studies at the Catholic Gymnasium in Paderborn, Weierstrass regularly
read Crelle’s Journal1 and decided to devote himself to Mathematics. But in 1834,
when he was 19 years old, his domineering father, who was a tax inspector, sent him
to the University of Bonn to study finance and laws.
But not having any interest for these studies, in Bonn he spend time in amusement,
and also reading Laplace’s “Mécanique céleste” and Jacobi’s work “Fundamenta
nova theoriae functionum ellipticarum”, on a topic that was very popular at the
time. This work, based of the recently published “Trait des fonctions elliptiques” by
Legendre, proved difficult for him. To learn the techniques and methods in elliptic
functions, he read a transcription of a lecture on modular functions by Christoph
Gudermann, a former student of Gauss that later became Weierstrass’ advisor.
As a result, four years later, he left the University with no degree, and the conflict between his academic duties and mathematical inclinations led to mental and
physical strain.
1Crelle
is the comon name of the “Journal für die reine und angewandte Mathematik”. Founded
by August Crelle in 1826 and edited by him until his death in 1855, it was the first major mathematical journal that was not a proceedings of an academy. Weierstrass and Kronecker were the
editors from 1881 to 1888.
1
2
JOAN CERDÀ
In 1839 he accepted to study at the Theological and Philosophical Academy at
Münster in order to become a secondary school teacher. One reason of his acceptance
was that Gudermann was teaching there.
Weierstrass attended Gudermann’s lectures on elliptic functions x = ϕ(α), doubly
periodic functions when extended on complex plane, which are the inverse of the
elliptic integrals,
Z x
4
X
du
p
α=
;
P (u) =
ak uk .
P (u)
0
k=0
These functions constituted the subject of his first paper, a study dated in 1840.
The starting point was Abel’s claim that elliptic functions should be quotients of
convergent power series.
Already being a Gymnasium teacher, completely isolated and without any contact
with any mathematician for discussions and with no acces to a mathematical library,
Weierstrass intensely worked at every free time for almost fourteen years, without
any knowledge of Cauchy’s work on complex functions.
He developed the theory of abelian functions, obtained by inversion of abelian
integrals
Z x
p
R(u, f (u))du,
α=
0
where R(u, v) is a rational function and f a general function, including polynomials,
exactly in the same way as elliptic functions are obtained from elliptic integrals.
To establish the foundations of his work, Weierstrass intensely worked on analysis without any
knowledge of Cauchy’s related results. So, in 1841,
two years before Laurent, Weierstrass established
in [W5] the Laurent expansion of a function and
obtained the Cauchy integral theorem for an annulus, and in [W4] he proved the Cauchy inequalities for the coefficients of a Laurent series, and his
preparation theorem for double series. This work
is the starting point of the Weierstrass analytical
methods for the theory of functions of one or several variables based on their expansion in power
series.
Weierstrass
The 1842 paper [W6], where Weierstrass proved
that a system of differential equations could be solved by a system of convergent
power series satisfying prescribed initial conditions, completed this work. There he
also showed how a convergent power series could be analytically continued outside
the convergence disk.
But this work, written when Weierstrass was still under the influence of Gudermann, who strongly encouraged him in his studies, remained unpublished until 1894,
when they were included in his Mathematische Werke, vol 1. It had no influence on
the development of mathematics of those years, but there, since 1857, he based his
theory of functions of complex variables.
His effort of having intensively working in mathematics, that Weierstrass had to
combine with his teaching duties at the Gymnasium, having to taught German,
botany, geography, history, gymnastics, and even calligraphy, in addition to mathematics and physics, and the conflicts that he had to endure as a student, were
WEIERSTRASS AND UNIFORM APPROXIMATION
3
probably the reasons why, around 1850, he began to suffer severe headaches and
convulsions.
Weierstrass came out of anonymity when in 1854 and 1856 he published in Crelle’s
two papers, that, later, Hilbert (1862-1943) considered the greatest achievement in
analysis. The first article [W1] concerned abelian functions and the second one [W2]
contained a complete version of his theory on hyperelliptic integrals,
αk =
p−1 Z
X
j=0
0
xj
y k dy
p
P (y)
(0 ≤ k < p; degree P = 2p + 1 or 2p + 2),
solving in a very original way Jacobi’s problem on the inversion of these integrals.
He then received several offers and accepted a professor position at the University
of Berlin, in 1856. There, his courses on applications of Fourier series and integrals
to mathematical physics, the introduction to the theory of analytic functions, elliptic functions, and applications to problems in geometry and mechanics, attracted
students from all over the world.
At Berlin he met two great mathematicians: Ernst Eduard Kummer (1810-1893)
and Leopold Kronecker (1823-1891), who was a close friend of Weierstrass for many
years. The three of them gave their university the reputation of being the best place
at which to study advanced mathematics.
Weierstrass acquired masterly skill in lecturing, his reputation attracted students
from around the world, and eventually some 250 students attended his classes. Over
the years, he organized his courses cyclicaly on “Introduction to the Theory of Analytic Functions”, “Theory of Elliptic Functions?, “Application of Elliptic Functions
to problems in Geometry and Mechanics”, “Theory of Abelian Functions”, “Application of Abelian Functions to the Solution of Selected Geometric Problems”, and
“Calculus of Variations”.
For a clear presentation of his courses, he built a solid construction for his mathematics. In his 1859-60 lectures about the introduction to analysis, he established
for the first time the foundations of this subject.
But in 1861 his health suffered a serious deterioration and took two years to
partially recover. Since then he needed to lecture sitting down on a chair with
a student helping him at the blackboard. Then headaches and convulsions were
replaced by recurrent attacks of bronchitis and phlebitis.
In his four-month 1863/64 course on analytic functions, Weierstrass started the
formulation of his theory of real numbers defined as the sums of convergent series of
rational numbers. There he proved the fact that complex numbers are the unique
commutative algebraic extension of real numbers, announced in 1831 by Gauss but
never proved. Until 1890 Weierstrass revised and extended these courses.
Two subjects centered Weierstrass activity: his program of basing analysis on
a firm foundation starting from a precise construction of real numbers, known as
arithmetization of analysis, and his work on power series and analytic functions.
In a 1875 letter to H. A. Schwarz where he criticized Riemann, Weierstrass states
“The more I think about the principles of function theory and I do this without
cease, the firmer becomes my conviction that this must be based on the foundation
of algebraic truths”.
The same year, in a letter to Kowaleskaya, he explained that what he asks to a
scientific work is unity of method, the sequential tracking of a definite plan and the
4
JOAN CERDÀ
appropriate working of details. He considered very important the attention to details and, although he preferred constructive arguments, he also admitted existence
proofs.
Weierstrass was not first in trying to give rigor to analysis. Bolzano, already
in 1817, and Cauchy had clear definitions of the concepts of limit, sum of series,
continuity of functions, derivatives, etc. But their definitions were not useful for
proofs.
For instance, Cauchy definition of limit was: “When the values attributed to a
variable approach indefinitely a fixed value, so come to differ as little as we want it,
the latter is called the limit of others”. Based on this definition, in his 1821 “Cours
d’analyse” de 1821, Cauchy gave the famous incorrect proof of the continuity of any
function which is the pointwise limit of continuous functions.
The Weierstrass rigor, with precise definitions of the basic concepts in terms of
inequalities that allow correct proofs, such our ε–δ definition of continuity and limit,
and with his work Weierstrass made mathematicians to be more precise when dealing
with mathematical analysis and definitely affected the evolution of mathematics.
But his methods and results were not easily accepted by the whole mathematical
community. His subtle arguments, including proofs of existence and contradiction,
were controversial. Weierstrass adopted the theory of Cantor, used for example to
prove the Bolzano-Weierstrass theorem on the existence of accumulation point for
infinite bounded subsets of R. In 1877, Kronecker, a very close friend of him, did not
accepted the actual infinite and openly criticized Cantor and Weierstrass in front of
the students, which meant his final distancing.
Concerning the topic of series, Weierstrass said that his work in analysis was
nothing else that power series. A general vision of his work on analytic functions is
included in [Bi].
2. Weierstrass and Riemann
Weierstrass’ theory on analytic functions was motivated by his study of elliptic and Abelian functions.
This study, guided by his intention to complete the work
started by Abel and Jacobi, would remain as a favorite
theme throughout his life and as a leitmotif of many of
the results presented in his courses.
In 1856 Weierstrass could not prove that every Abelian
function was the quotient of two sums of power series:
“As far as I know, here there is a problem that has not
yet been studied in general, but this is nevertheless of
particular importance in the theory of functions”.
Riemann
A continuation of Weierstrass’ paper never appeared
and, in 1857, Riemann2 presented a new treatment of Abelian integrals in the fourth
2Bernhard
Riemann (1826-1866), Riemann, who was very shy and with great fear to speak in
public, from a young age showed great math skills. At age 19 he began studies to become a priest
as his father, who in the same year allowed him to study mathematics at Gttingen, where Riemann
had Gauss as a teacher in an elementary course.
But Göttingenn wasn’t still the best place to do mathematics and, in 1847, Riemann went on
to the University of Berlin to study with Jacobi, Dirichlet, Steiner, and Eisenstein.
Dirichlet had a strong influence on him. In order to avoid long calculations as much as possible,
he based his ideas on intuitive basis, to extract very precise analysis. Riemann adopted these
WEIERSTRASS AND UNIFORM APPROXIMATION
5
article [Ri] of a series of four in volume 54 of Crelle, Riemann including a summary of
his geometric approach to the theory of functions of a complex variable contained in
the thesis. There, to treat multi-valued functions and their integrals, he introduced
the idea of representing the branches of a function by a surface multiply covering
the plane.
Riemann considered the Weierstrass work as a special case of his own and that a
continuation of the “nice results” of the 1856 paper would show “how much their
results and their methods coincided”. After the publication of Riemann’s paper,
Weierstrass decided to withdraw the continuation of his article submitted to the
Berlin Academy.
The theory of complex functions by Riemann became a reference in Weierstrass’
courses. As a reply to the results of Riemann, for twenty years he devoted himself
to a deep study of analytic functions that should establish the foundations of the
theory of elliptic and Abelian functions.
The personal relationship between Weierstrass and Riemann was friendly. Weierstrass deeply admired Riemann. He said that he had loved Riemann as a brother
and often told Mittag-Leffler that he was an “anima candida come je n’en ai jamais
connu”
Concerning mathematics, there was a great mutual influence, but their methods
were very different. Riemann did not hesitate to use results of other people, but
Weierstrass usually admitted them only after a careful reworking.
Riemann based his theory of analytic functions on harmonic functions,3 that is,
functions with given boundary values that minimize the Dirichlet integral
Z
D(u) :=
|∇u(x)|2 dx,
Ω
and he called Dirichlet principle this variational property.
We notice that, if u ∈ C(Ω̄) ∩ C 2 (Ω) is a solution of the RDirichlet problem
4u = 0
R
on Ω on u = f on ∂Ω, partial integration gives 0 = − Ω ϕ4u = Ω ∇ϕ · ∇u for
methods, and it was in Berlin where he built his general theory of complex variables that would
be the basis of the most important part of his work.
In 1849 he returned to Göttingen, where in 1851, supervised by Gauss, he presented the thesis
studying functions of a complex variable and, in particular, what we call Riemann surfaces. There
he used the Dirichlet Principle, who had learned from Dirichlet in Berlin.
In 1853, Gauss asked him to prepare a Habilitationsschrift. He did it for one year, working on
the representation of functions by trigonometric series, including his study on the integral. For
the Habilitation he had to talk about a topic and he prepared three. Two were about electricity
and one about geometry, which was chosen by Gauss. His presentation, in 1854, was received with
enthusiasm by the audience.
In 1855 Gauss was succeeded by Dirichlet in his Professorship in Gttingen, and Riemann occupied this chair in 1859, after Dirichlet’s death. That same year, at the proposal of Weierstrass,
Kummer and Borchardt, three mathematicians of Berlin, Riemann was elected member of the
Academy. It was in his introduction to the Academy where he presented the famous Riemann
hypothesis.
Riemann died in 1866 of tuberculosis in the Italian town of Selasca, where he went to fight
against the disease that he caught just after his marriage, in 1862.
3In Fourier analysis, functions on the circle are described by their harmonics. Similarly, the
functions on the sphere are described in terms of polynomials with a null Laplacian and were
called harmonic polynomials by Thompson (Lord Kelvin) and Peter Tait in his “Treatise of Natural
Philosophy” (1879). Since the early twentieth century, the term “harmonic” is also applied to all
functions satisfying the Laplace equation.
6
JOAN CERDÀ
every ϕ ∈ Cc∞ (Ω); hence
Z
Z
Z
2
2
|∇u| + |∇ϕ| =
|∇(u + ϕ)|2 = D(u + ϕ),
D(u) ≤
Ω
Ω
Ω
and u is a minimizer for the Dirichlet integral.
Conversely, if u is a minimizer for D(u) in the family of all u ∈ C(Ω̄) ∩ C 2 (Ω) such
that u = f on ∂Ω, from
Z
Z
Z
Z
2
2
2
2
|∇(u + tϕ)| ≥
|∇u| ⇔ t
|∇ϕ| − 2t ∇ϕ · ∇u ≥ 0
Ω
Ω
Ω
Ω
we obtain
Z
Z
ϕ4u = −
Ω
∇ϕ · ∇u = 0
ϕ ∈ Cc∞ (Ω) ,
Ω
which is equivalent to 4u = 0.
The existence of minimizers in variational problems was generally accepted since
the work by Gauss on the Newtonian potential and by Dirichlet on the electrostatic
potential, based on physical arguments, and the Dirichlet principle was the source
of many important results.
But in Weierstrass’ opinion the existence of the minimizer function u was not
clear, and, although Riemann accepted that some of his proofs were incomplete, he
was convinced of the existence of a minimum for the Dirichlet integrals.
After Riemann’s death, Weierstrass often criticized his methods. In 1870 he presented to the Academy of Sciences [W3] a famous argument against the Dirichlet
Principle. It was the example of one variable
Z 1
x2 ϕ0 (x)2 dx
D(ϕ) =
−1
on A = {ϕ ∈ C 1 [−1, 1]; ϕ(−1) = a, ϕ(1) = b}, with a 6= b. Then inf ϕ∈A D(ϕ) = 0,
since D(ϕn ) ≥ 0 becomes arbitrary close to 0 if
b + a b − a arctan(nx)
+
,
2
2
arctan n
but D(ϕ) > 0 if ϕ ∈ A, because D(ϕ) = 0 would imply ϕ0 = 0 and ϕ = C constant,
in contradiction with a 6= b.
Hence, the Dirichlet principle was unacceptable in the form it was understood,
and one of the fundamental facts of Riemann’s function theory was questioned. His
method was abandoned by many mathematicians, who tried to prove his results by
using other methods.4
Moreover Weierstrass also observed that apparently Riemann thought that an
analytic function can be continued along any curve that avoids critical points. He
remarked
this is not possible in general, as shown by the lacunar series u(z) =
P∞ n athat
n
n=0 b z , where a is an odd integer and 0 < b < 1 such that ab > 1 + 3π/2. In
this case, the circle |z| = 1 is the natural boundary for the function u.
ϕn (x) :=
4In
1899, Hilbert established the validity of Dirichlet’s principle under strong conditions on the
domain Ω and on the class A of admissible functions, and he predicted that a final solution would
be found by weakening the conditions. In 1933 Zaremba and Nikodym solved this problem. We
refer to [Mo] for a detailed description of the history of the Dirichlet problem.
Now Riemann’s method can be used to prove his theorem of conformal representation. See
e.g. [Cy].
WEIERSTRASS AND UNIFORM APPROXIMATION
7
Indeed, the convergence radius of the series is 1, and the real part of the sum on
z = eiϑ is the periodic function
f (ϑ) =
∞
X
bn cos(an ϑ),
(1)
n=0
the famous pathological continuous function with no derivative at any point, discovered by Weierstrass in 1862 (published in 1872). This was a counterexample to
the general belief that continuous functions had to be differentiable, except at some
special points.
Weierstrass also remarked that Riemann had stated without a proof in 1861 that
f (x) =
∞
X
sin(n2 x)
n=1
n2
,
(2)
was an example of a nowhere differentiable continuous function, and that “it is
somewhat difficult to check that it has this property”. Much later, in 1916, Hardy
proved that this function has no derivative at any point except at rational multiples
of π, for rational numbers that can be written in reduced form as p/q with p and q
odd integers.
Many of the Weierstrass results were motivated by his criticism of Riemann methods and his “geometric fantasies”. He chose the way of power series by his conviction
that it was necessary to construct the theory of analytic functions on simple “algebraic truths”.
When comparing his construction and those by Cauchy and Riemann, based on
the Cauchy-Riemann differential equations, Weierstrass indicates that in fact “to
prove the possibility of analytic continuation, Riemann had to use power series, a
fact that does not appear in his complete works, but that he included in his courses”.
Despite his criticisms, Weierstrass firmly believed in the validity of the results of
Riemann. For example, he asked Hermann Schwarz to try to prove the existence
theorems without using Dirichlet’s Principle, a proof that Schwarz achieved in 1869–
70.
In [P1] and [P2], Poincaré defined Weierstrass as a logician, reducing everything
to the consideration of series and analytic transforms, and said that we can not read
his work to find a single image, while Riemann is an intuitive, using geometry, and
each of his discoveries is an image that can not be forgotten when its meaning is
understood.
But this is an oversimplification, as noted Hadamard, who did not share Poincaré’s
opinion, and says in [Ha] that Weierstrass always had an initial intuition. He remarks
that actually there is one picture in the important work where Weierstrass presents
his calculus of variations. Hadamard observes that, after an initial step, a look on
the picture allows one to reconstruct all the logical steps of the proof. He also notes
that, on the other hand, the geometric elements play no role in the intuition that
appears on Riemann’s work on prime numbers.
Weierstrass published little, probably because of his critic sense and of the precariousness of his health. Much of the work presented in his lectures was known
much later from the notes of his courses and from publications of his students, and
were included in his collected works in seven volumes that he decided to supervise.
8
JOAN CERDÀ
The two first volumes appeared in 1894 and 1895. The other five were published
between 1903 and 1927, after his death of pneumonia in 1897. He spent his last
three years confined in a wheelchair.
For a more detailed description of the interactions between Weierstrass and Riemann, a good reference is [Ne].
3. Uniform convergence
“... il y avait tout un type de raisonnements
qui se ressembaient tous et qu’on retrouvait
partout; ils étaint parfaitement rigoureux,
mais ils étaient longs. Un jour on a imaginé
le mot d’uniformité de la convergence et ce
mot seul les a rendus inutiles...”
Poincaré in “Science et Méthode” [P1]:
The (pointwise) convergence of a sequence of functions was used in a more or less
conscious way from the beginning of the infinitesimal calculus and, as we have seen,
the continuity of a function was clearly defined by Bolzano and Cauchy. Cauchy did
not notice the uniform convergence, and, in 1821, he believed to have proved the
continuity of limits and sums of series of continuous functions.
Dirichlet found a mistake in Cauchy’s proof,5 and Fourier and Abel presented
counter examples using trigonometric series. Abel then proved the continuity of the
sum of a power series by means of an argument using the uniform continuity in that
special case.
The concept of uniform convergence probably appears for the first time in a 1838
article by Gudermann about elliptic functions, where he talks about “convergence
in a uniform way” when the “mode of convergence” does not depend on the values
of the variables. He refers to this convergence as a “remarkable fact”, but, without
giving a formal definition or using it in proofs.
This convergence was also considered in 1847 by Seidel in a criticism to Cauchy
(talking without a definition about slow convergence as the absence of uniform convergence) and, independently, by Stokes in 1849 (about infinitely slow convergence,
with no reference to Cauchy), but without having any impact on further development.
It was Weierstrass, in a work in Münster dated 1841 but only published in 1894, after attending Gudermann’s course, that introduced the term “uniform convergence”
and used it with precision.
G. H. Hardy, when comparing the definitions by Weierstrass, Stokes and Seidel in
the paper “Sir George Stokes and the concept of uniform convergence”, remarked
that “Weierstrass’s discovery was the earliest, and he alone fully realized its farreaching importance as one of the fundamental ideas of analysis”.
At the end of the nineteenth century, by the influence of Weierstrass and Riemann,
the application of the notion of uniform convergence was developed by many authors
such as Hankel and du Bois-Raymond in Germany, and Dini and Arzelà in Italy.
5Dirichlet
also found a lack on the definition of the integral of a continuous function by Cauchy
and defined the notion of uniform continuity.
WEIERSTRASS AND UNIFORM APPROXIMATION
9
For instance, Ulisse Dini proved the result named after him stating that if an
increasing sequence of continuous functions fn : [a, b] → R is pointwise convergent,
the convergence is uniform.
Uniform convergence was used by Weierstrass in his Example (1) showing that
the uniform limit of differentiable functions need not be differentiable. As we have
seen, this was also Riemann’s conviction with Example (2).6 We will briefly be
concerned with the Weierstrass approximation theorem, that in fact is the converse:
every continuous function is the uniform limit of differentiable functions, in fact the
uniform limit of polynomials.
M. Kline, in [Kl], explains that in 1893 Hermite said to Stieltjes in a letter: “Je
recule de terreur et d’aversion devant ce mal déplorable que constituent les fonctions
continues sans dérivées”, and this was a very extended opinion. In this sense, talking
about teaching Mathematics, Poincaré says in [P1] and in [P3]:
“La logique parfois engendre des monstres. Depuis un demi-siècle on a vu surgir
une foule de fonctions bizarres qui semblent s’efforcer de ressembler aussi peu que
possible aux honnêtes fonctions qui servent à quelque chose. Plus de continuité, ou
bien de la continuité, mais pas de dérivées, etc. Bien plus, au point de vue logique,
ce sont ces fonctions étranges qui sont les plus générales, celles qu’on rencontre sans
les avoir cherchées n’apparaissent plus que comme un cas particulier. Il ne leur reste
qu’un tout petit coin.
Autrefois, quand on inventait une fonction nouvelle, c’était en vue de quelque
but pratique; aujourd’hui, on les invente tout exprès pour mettre en défaut les
raisonnements de nos pères, et on n’en tirera jamais que cela.”
Now we are familiar with continuous nowhere differentiable functions and they
are needed to deal with fractals, Brownian motion, wavelets, chaos, etc., but these
examples were a shock for the mathematical community of the 19th century.
4. 1885: The Weierstrass approximation theorem
His 1885 article “Über die analytische Darstellbarkeit sogenannter willkürlicher
Funktionen einer reellen Veränderlichen” [KW], divided in two parts, has to be
included in the Weierstrass program of representing functions by means of power
series. There he proved the following approximation theorem when he was aged 70.
THEOREM 1. If f ∈ C[a, b] is a continuous real function and ε > 0, then
|f − p| ≤ ε for some polynomial p.
The interest of this result was immediately appreciated and one year later a
translation, also in two parts, was included in the Journal de Liouville.7
Many of the best mathematicians became strongly interested in this result and
gave new demonstrations and new applications. Among them are Runge, Lerch and
Mittag-Leffler, Weierstrass students, and many others, as Picard, Fejér, Landau, de
la Vallée Poussin, Phragmén, Lebesgue, Volterra, Borel and Bernstein (cf. e.g. [Pi].)
6In
the 1830’s, Bolzano also constructed a continuous nowhere differentiable function that was
the uniform limit of certain piecewise linear functions, but with the usual mistake of assuming that
pointwise limits of continuous functions are also continuous. He only stated that his function was
not differentiable in a dense set of point, but in fact the function is everywhere non-differentiable.
7“Journal de Liouville” was the name given to the french “Journal de Mathématiques Pures
et Appliquées”, founded in 1836 by Liouville. It is the older one, after the german “Journal de
Crelle”, founded ten years before.
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JOAN CERDÀ
In order to describe the Weierstrass paper, and several of the proofs that appeared
later, if needed we can suppose [a, b] = [0, 1], f (0) = f (1) = 0 and |f | ≤ 1, without
loss of generality.
Indeed, to change the domain we observe that, if Pn (t) → f (αt + β) uniformly
on [0, 1], then also Qn (x) = Pn ((β − x)/α) → f (x) uniformly on [a, b]. To obtain
f (0) = f (1) = 0, change f (x) by f (x) − αx − β, and then Pn (x) → f (x) ⇒
Pn (x) + αx + β → f (x). Finally, we divide f by an upper bound M of |f | and it
follows from Pn → f /M , with |f /M | ≤ 1, that M Pn → f .
The basic elements of THEOREM 1 are included in the first part of [KW], in
Theorems (A), (B) and (C). This part starts with the consideration of the function
Z ∞
u−x 2
1
F (x, t) = √
f (u)e−( t ) du,
t π −∞
similar to the solution of the heat equation ∂t F (x, t) = ∂x2 F (x, t) on the half plane
t > 0 with initial values F (x, 0) = f (x), given by Fourier and Poisson.
WEIERSTRASS AND UNIFORM APPROXIMATION
11
The proof is divided in two parts. First it is shown that F (x, t) → f (x) uniformly
on every interval of R as t ↓ 0, and then that, for a given t > 0, F (·, t) is an entire
function that can be approximated by polynomials using the partial sums of the
Laurent series, as follows.
Theorem (A) Let f be a [bounded] continuous function on R. Then there are many
ways to choose a family of entire functions F (x, t), with t > 0 a real parameter, such
that limt→0 F (x, t) = f (x) for every x ∈ R.
Moreover, the convergence is uniform on every interval [x1 , x2 ].
In fact, Weierstrass uses the convolution
Z
1 ∞
u − x
F (x, t) =
du,
f (u)ψ
t −∞
t
where ψ is a general kernel satisfying the conditions
(a) ψ is similar to f (bounded and continuous).
(b) ψ ≥ 0 and ψ(−x) = ψ(x),
R∞
(c) The improper integral −∞ ψ(x) dx is convergent (Weierstrass divides by ω =
R∞
R∞
ψ(x) dx and the one can suppose that −∞ ψ(x) dx = 1).
−∞
step,
by Cauchy’s criterion, he shows that, for every x, the integrals
R b In a firstu−x
f
(u)ψ
du converge as a, b → ∞ using the average property of the integrals
t
−a
and the properties of ψ:
Z
Z
1 a2
u − x
u − x
1 b2
du −
du
f (u)ψ
f (u)ψ
∆ :=
t −b1
t
t −a1
t
Z
Z
u − x
u − x
1 −a1
1 b2
f (u)ψ
f (u)ψ
=
du −
du
t −b1
t
t −a2
t
Z (b2 −x)/t
Z (b1 +x)/t
ψ(v) dv
ψ(v) dv + f (ξ2 )
= f (ξ1 )
(a2 −x)/t
(a1 +x)/t
R (b2 −x)/t
R (b1 +x)/t
and, from |f | ≤ 1, |∆| ≤ M (a1 +x)/t ψ(v) dv + M (a2 −x)/t ψ(v) dv → 0 if b1 > a1 →
∞ and b2 > a2 → ∞.
Then, as we currently do with summability kernels, he decomposes F :
Z x+δ o
Z +∞ Z x
Z
1 n x−δ
u − x
F (x, t) =
f (u)ψ
+
+
+
du
t
t
x+δ
x−δ
x
−∞
Z δ
Z ∞
t
= (f (ξ1 ) + f (ξ2 ))
ψ(u) du + [f (x − tu) + f (x + tu)]ψ(u) du,
δ
t
0
and F (x, t) − f (x) is equal to
Z ∞
Z
(f (ξ1 ) − f (x))
ψ(u) du + (f (ξ2 ) − f (x))
δ
t
∞
ψ(u) du
δ
t
Z
+
δ
t
[f (x − tu) + f (x + tu) − 2f (x)]ψ(u) du.
0
After taking absolute values and using the uniform continuity of f on every interval
[x1 , x2 ], it follows that the last integral is bounded by any ε > 0 if δ is small, and
tR ≤ t0 , for some t0 > 0. The two first terms of the right hand side are bounded by
∞
ψ(u) du, which also is ≤ ε if t ↓ 0, for any δ.
δ/t
12
JOAN CERDÀ
Weierstrass explicitly observes that the convergence is uniform on [x1 , x2 ].
Theorem (B) If f is as in Theorem (A) and ε > 0, then there are many ways of
choosing a polynomial G such as |f (x) − G(x)| ≤ ε for every x ∈ [x1 , x2 ].
In the proof he observes that for many kernels, as in the case of the heat kernel
2
W (x) = (1/π)e−x (that we call Weierstrass kernel), every F (·, t) is entire and can
be represented
∞
X
F (x, t) =
A n xn .
n=0
Hence, for any ε > 0, there is a polynomial GN (x) =
|F (x, t) − GN (x)| ≤ ε
PN
n=0
An xn which satisfies
(x ∈ [x1 , x2 ]).
That is, |f − GN | ≤ 2ε.
The approximation has been obtained, but Weierstrass wants to arrive at a representation of f by a series, and he proves a third theorem:
Theorem (C) If f (x) is as before, it can be “represented” in many way as [the
sum of] a series of polynomials which is uniformly convergent on every interval and
absolutely convergent for every x.
P
The proof is easy. He writes ε =
εn and approximates f as in Theorem (B) by
Taylor polynomials Gn so that
|f
−
G
n | < εn on [−an , an ], with an ↑ ∞. Finally the
P∞
la telescopical series G1 + n=1 (Gn+1 − Gn ) satisfies the theorem.
In the case of a continuous function f on an interval [a, b], Weierstrass extends f
to R defining f (x) = f (a) if x < a and f (x) = f (b) if x > b. The sequence {Gn } of
polynomials tends to f , uniformly on [a, b].
In this second part, although du Bois-Reymond had built a continuous function
with Fourier series that does not converge to the function in a dense set of points,
using methods of complex variable, Weierstrass shows that every periodic continuous
function is the uniform limit of a sequence of trigonometrical polynomials:
THEOREM 2. Let f ∈ C2π (R), a continuous 2π-periodic function on R, and ε > 0.
Then there are trigonometric polynomials t such that |f − t| ≤ ε on R.
2
The proof goes as follows: If ψ(z) = (1/π)e−z (or similar), then, for every t > 0,
Z
u − z 1 +∞
Ft (z) :=
f (u)ψ
du
t −∞
t
is an entire periodical function, so that
Gt (z) := Ft
log z i
is an univalent analytic function, well
on R
P defined non C \ {0}, which is real valued
ix
and has a Laurent series Gt (z) = +∞
c
z
that
on
the
unit
disc
z
=
e
is
k=−∞ t,k
Ft (x) =
+∞
X
k=−∞
with uniform convergence.
ct,k eikx ,
WEIERSTRASS AND UNIFORM APPROXIMATION
13
A two-ε argument completes the proof: If t > 0 is small and N large, then
+N
X
ikx |f (x) − Ft (x)| ≤ ε, Ft (x) −
ct,k e ≤ ε
(∀ x).
k=−N
In fact we can suppose that
P+N
k=−N
ct,k eikx is real, since
+N
+N
X
X
ikx ct,k e ≤ Ft (x) −
ct,k eikx .
Ft (x) − <
k=−N
k=−N
Weierstrass observes that this result justifies the solution of the heat equation
given by Fourier for a thin annulus with a given initial temperature, since his trigonometric polynomials also satisfy the heat equation.
5. 1891: Picard and the approximation obtained from the Poisson
equation
Émile Picard (1856–1941) 8 obtained in 1891 [Pc] a new proof similar to the one
given by Weierstrass changing the heat kernel by the Poisson kernel, used to solve
the Dirichlet problem corresponding to a stationary distribution of the temperature
u(r, ϑ) on the disc D = {(r, ϑ); 0 ≤ r < 1}, for a continuous and periodic function
f (ϑ) = u(r, ϑ) representing the temperature on the boundary.
If z = x + iy = reiϑ ∈ D, then the solution is given by the function
Z π
Z π it
1 − r2
1
e +z
1
f (t)
dt = <
f (t) dt,
u(z) =
2
2π −π
1 − 2r cos(ϑ − t) + r
2π −π eit − z
which is harmonic, as the real part of a holomorphic function. That is, 4u(x, y) = 0.
On the other hand, if 0 < r < 1, the summation of a geometric series gives
1
1 − r2
= Pr (ϑ − t)
2π 1 − 2r cos(ϑ − t) + r2
with
Pr (s) =
∞
1 X |k| iks
r e .
2π k=−∞
The family of periodic functions {Pr }0<r<1 , currently called the Poisson kernel,
satisfies
8Picard
was a brilliant student who initially disliked Mathematics, but after completing secondary studies, he became fascinated when reading an algebra book.
He had the opportunity of visiting Pasteur, who spoke about pure science and persuaded Picard,
who, having to choose between the École Polytechnique which, in principle, prepared one to be an
engineer, and the École Normale Supérieure, with its pure scientific orientation, chose the latter.
He had Darboux as his advisor, and Bernstein, Hadamard, Julia, Painlevé and A. Weil, and
many others, were his students. He made important contributions in the theory of functions and
in differential equations, with the use of successive approximations, and he extended properties of
the Laplace equations to more general elliptic equations. He also introduced the Picard group in
the theory of algebraic surfaces.
His famous “Traité d’analyse” in three volumes, first published between 1891 and 1896 and
revised with each subsequent edition, was followed by many students who could enjoy his lucid
style and clear exposition, presenting specific examples before the general theory. He included
several proofs of the Weierstrass theorem, but without giving a reference of its origin.
Hadamard wrote in 1941: “A striking feature of Picard’s scientific personality was the perfection
of his teaching, one of the most marvellous, if not the most marvellous, that I have ever known.”
14
JOAN CERDÀ
(a) Pr ≥ 0,
(b) RPr (−s) = Pr (s),
π
(c) −π Pr (s) ds = 1, and
(d) sup0<δ≤|t|≤π Pr (t) ≤ Pr (δ) → 0 as δ ↓ 0.
As a consequence, as it happens with the Weierstrass kernel,
Z π
iϑ
u(re ) =
f (t)Pr (ϑ − t) dt → f (ϑ)
−π
uniformly, as r ↑ 1.
If 0 < r < 1, then
1
u(re ) =
2π
iϑ
Z
π
∞
X
r|k| eik(ϑ−t) f (t) dt =
−π k=−∞
∞
X
ck r|k| eikϑ
k=−∞
with
Z π
1
f (t)e−ikt dt,
ck =
2π −π
and the series is uniformly convergent, by the Weierstrass M -test.
If f : [0, 1] → R is a continuous function such that |f | ≤ 1
and f (0) = f (1) = 0, then it is extended by zero to [−π, π]
and to a periodic function to R. For every ε > 0, there is
some 0 < r < 1 such that
sup |f (ϑ) − u(reiϑ )| ≤ ε.
−π≤ϑ≤π
But
∞
X
|k|≥N
|ck r|k| eikϑ | ≤
∞
X
r|k| = 2
|k|≥N
rN
1−r
and one can choose N so that 2rN /(1 − r) < ε.
By adding these inequalities,
X
rN
≤ 2ε.
sup |f (ϑ) −
ck r|k| eikϑ | ≤ sup |f (ϑ) − u(reiϑ )| + 2
1−r
−π≤ϑ≤π
−π≤ϑ≤π
Picard
|k|<N
In this way Picard obtains a trigonometric polynomial Q(ϑ) which uniformly
approximates f , and in turn Q is approximated by a Taylor polynomial.9 This is
the proof presented, by instance, in [Se].
At the end of his paper, Picard observes that the same method gives the approximation theorem for functions of several variables. He also says that his proof is
based in an inequality due to H. Schwarz; in fact, in [Sc] we almost find the proof.
9Instead
of this last approximation by a Taylor polynomial, similar to the one given by Weierstrass , in 1918 and following an idea from Bernstein, de la Vallée Poussin gave a new an more
direct approximation:
For f ∈ C[−1, 1] such that f (−1) = f (1), the even function
g(ϑ) := f (cos ϑ)
(|ϑ| ≤ π)
is approximated by a 2π-periodic trigonometric polynomial t, |g − t| ≤ ε, which in turn decomposes
PN
in its even and odd parts, t = te + to (te (ϑ) = (t(ϑ) + t(−ϑ))/2 = k=0 ak cos(km)). Since g is
even, it follows that |g − te | ≤ ε.
Every cos(kϑ) is a polynomial with degree k of cos ϑ, cos(kϑ) = Tk (cos ϑ (Tk is a Txebisxef
PN
polynomial) and p(x) = k=0 ak Tk (x) satisfies |f − p| ≤ ε.
WEIERSTRASS AND UNIFORM APPROXIMATION
15
When the Weierstrass Mathematische Werke were reedited in 1903, the same remark about the case of several variables was included. Probably Weierstrass directed
this edition, necessarily before 1897.
6. 1898: Lebesgue and polygonal approximations
New proofs of the Weierstrass theorem start from a first approximation of the
continuous function f : [0, 1] → R by a polygonal function g on nodes 0 = x0 <
x1 < · · · < xm = 1,
g(x) = g1 (x)+[g2 (x) − g1 (x)]χ(x − x1 ) + [g3 (x) − g2 (x)]χ(x − x2 ) + · · ·
+ [gm (x) − gm−1 (x)]χ(x − xm−1 )
(3)
with χ(x) = 1 if x ≥ 0 and = 0 if x < 0, and where gj is the line connecting
(xj−1 , f (xj−1 )) with (xj , f (xj )),
gj (x) = f (xj−1 ) +
x − xj−1
[f (xj ) − f (xj−1 )].
xj − xj−1
(4)
If we denote x+ = max(x, 0) = (|x| + x)/2 and g1 (x) = cx + c0 , then also
g(x) = cx + c0 + c1 (x − x1 )+ + · · · + cm−1 (x − xm−1 )+ ,
or, since (x − xj )+ = (|x − xj | + x − xj )/2,
g(x) = ax + b0 + b1 |x − x1 | + · · · + bm−1 |x − xm−1 |.
(5)
This is how Lebesgue, in his first paper [L1], written when he was 23 years old,
gives one of the most elegant proofs of THEOREM 1, with the polygonal function
g written as in (5).
He observed that an approximation of |x| by a polynomial p was sufficient, since,
if
|x| − p(x) ≤ ε
(|x| ≤ 1),
then
|x − xk | − p(x − xk ) ≤ ε
(x ∈ [0, 1] ∩ [xk − 1, xk + 1]; k = 1, . . . , m − 1)
and an approximation of g by polynomials easily follows.
To obtain p(x), Lebesgue wrote
p
√
|x| = 1 − (1 − x2 ) = 1 − z,
with z = 1 − x2 , and then
√
1−z =
∞
X
n
C1/2
(−z)n ,
n=0
using the binomial formula with
− 1) · · · ( 12 − n + 1)
=
.
n!
From the Stirling formula he checked that the convergence radius was 1, and that
for |z| = 1 the convenient convergence holds.
n
C1/2
1 1
(
2 2
The delicate point in the
√ Lebesgue proof was precisely the discussion of the convergence of the series of 1 − z at |z| = 1. A simple way to overcome this difficulty
16
JOAN CERDÀ
is to use that (1 − δz)1/2 → (1 − z)1/2 uniformly as δ ↑ 1, and then it is easy to
approximate (1 − δz)1/2 (0 < δ < 1) by polynomials, since
1/2
(1 − δz)
=1−
∞
X
an δ n z n
n=1
uniformly on |z| < δ −1 , and δ −1 > 1.
But probably the most clever way to approximate |x| on [−1, 1] by polynomials
was obtained in 1949 by N. Bourbaki [Bo], who recursively defined a sequence of
polynomials pn starting from p0 = 0 and then
1
pn+1 (t) = pn (t) + (t − p2n (t)).
2
√
√
√
Since t − pn+1 (t) = ( t − pn (t))(1 − 12 ( t − pn (t))), it follows by induction that
√
0 ≤ pn (t) ≤ t, and {pn } is increasing on [0, 1]. By Dini’s theorem, pn → h
uniformly, with h ≥ 0 such that
1
h(t) = h(t) − (t − h2 (t)),
2
that is, h(t) =
√
t. Hence qn (x) = pn (x2 ) → |x| uniformly on [−1, 1].
In fact, this approximation of |x| is useful to prove the
Stone-Weierstrass theorem, an important important extension of the Weierstrass theorem. See e.g. [Bo], [Di] o [Or].
In 1908, referring to the proof by Weierstrass and Picard,
and to those by Fejér and Landau that we present below,
in a letter to Landau, Lebesgue observed that they must
be considered in a general setting of convolutions with sequences of nonnegative kernels that are approximations of
the identity, a remark that he developed in [L2].
Let us say that, yet in 1892, in an unnoticed paper in
Check
[Le], M. Lerch also proved THEOREM 1 using an
Lebesgue
approximation of the polygonal function g by a Fourier series
of cossinus
Z 1
∞
A0 X
+
An cos(nπx);
An = 2
g(t) cos(nπt) dt,
2
0
n=1
by the 1829 method of Dirichlet on the Fourier series
∞
A0 X
f∼
+
An cos(2πnx) + Bn sin(2πnx) ,
2
n=1
writing the Fourier sums as integrals,
N
A0 X
sN (f, x) =
+
An cos(2πnx) + Bn sin(2πnx)
2
n=1
Z 1/2
=
f (t)DN (x − t) dt,
−1/2
(6)
(7)
WEIERSTRASS AND UNIFORM APPROXIMATION
17
where
DN (t) = 1 + 2
N
X
cos(2πnt).
(8)
n=1
As proved by Heine in 1870 [He], sN (f, x) → f (x) uniformly if f is continuous and
piecewise monotone, or if it is piecewise differentiable, as in the case of g.
This is the proof contained in [CJ].
In 1897, V. Volterra (1856-1927) presented a similar proof of THEOREM 2.
7. 1901: Runge’s phenomenon
To obtain a direct proof of THEOREM 1, it is natural
to try a substitution of the polygonal approximation of our
continuous function f by an interpolation of the values of
(xj , f (xj )) by polynomials with increasing degrees, when the
number of nodes xj is also increasing.
But Carl Runge (1856 Bremen -1927 Göttingen),10 when
exploring the behavior of errors when using polynomial interpolation to approximate certain functions, discovered an
important numerical fact, showing that going to higher degrees does not always improve accuracy, similar to the Gibbs
phenomenon in Fourier series approximations.
Runge observed that if the function
Runge
1
f (x) =
1 + 25x2
on the interval [−1, 1] is interpolated at n + 1equidistant nodes xi ,
xj = −1 + (j − 1)
10Carl
2
n
(j = 1, 2, . . . , n + 1),
Runge (1856 Bremen -1927 Göttingen), after the adolescence spent in Havana, in 1876
enrolled at the University of Munich to study literature and philosophy. There he became close
friend of Max Planck and both of them went to Berlin in 1877, where, after attending Weierstrass’s
lectures, Runge turned to pure mathematics.
In 1880 he submitted to the University of Berlin his doctoral dissertation on differential geometry.
Although Weierstrass was his advisor, in fact he had not suggested the topic of the thesis; rather
this had come out from discussions with other students. However, Rung always considered himself
a student of Weierstrass.
After qualifying to be a Gymnasium teacher, in 1881 he returned to Berlin and joined the group
of Kronecker, where he worked on the numerical solution of algebraic equations whose roots could
be expressed as infinite series of rational functions of the coefficients. His Habilitation thesis,
submitted to Berlin in 1883, contained a numerical method that included the methods by Newton,
Bernoulli and Gräffe as special cases.
In 1886 he obtained a chair at Hannover, where he remained for 18 years. There, influenced by
Emil du Bois-Reymond, an older brother of the mathematician Paul du Bois-Reymond and one of
the most impressive professors at Berlin, Runge moved away from pure mathematics to study the
spectrum of elements other than hydrogen. In 1887 he married an Emil’s daughter.
It is in Hannover where in 1901 he wrote the paper [R3] on Runge’s phenomenon, and in 1904
he was appointed as Professor of Applied Mathematics, where he worked out many numerical and
graphical methods and gave numerical solutions of differential equations.
18
JOAN CERDÀ
by a polynomial pn of degree ≤ n, the resulting interpolation oscillates toward the
extremes of the interval and the error tends toward infinity when the degree of the
polynomial increases; that is,
lim
max |f (x) − pn (x)| = ∞.
n→∞
−1≤x≤1
It is natural to try interpolation with nodes more densely distributed toward the
edges of the interval, since then the oscillation decreases. In the case f : [−1, 1] → R,
choosing
−1 < xn < · · · < x1 < 1,
(xk = cos((2k − 1)π/2n)),
(9)
which are the zeroes of the Txebisxef polynomials, the error is minimized and decreases as the degree of the polynomial increases.
Runge’s example
But thirteen years later, in 1914, Faber [Fa] would show that, for any fixed triangular infinite matrix of nodes,
−1 ≤ xn,n < · · · < x0,n ≤ 1,
there is always a continuous function such that the corresponding sequence of interpolating polynomials pn of degree n (pn (xnj ) = f (xnj )) is divergent.
Finally, in 1916 Fejér [R1] proved that THEOREM 1 can be obtained by an
interpolation scheme typus Hermite, with nodes (9) and where the interpolation
polynomials are the polynomials Hn of degree ≤ 2n − 1 such that H(xk ) = f (xk )
and Hk0 (xk ) = 0.
It is worth noticing that in 1885 Runge also had proved the following result ([R1])
which is closely related with THEOREM 1: If K is a compact subset of C, f an
holomorphic function defined on a neighborhood of K, and A any set containing at
least one point of every “hole” (a bounded component) of C \ K, then there exists
rk → f uniformly on K, where every rk is a rational function with singularities
located in A.
Hence, if C \ K is connected, every rk is a polynomial.
WEIERSTRASS AND UNIFORM APPROXIMATION
19
The same year of the proof of THEOREM 1 by Weierstrass, Runge [R2] also
proved that every polygonal function g, with the la representation (3),
g(x) = g1 (x) +
m−1
X
[gj+1 (x) − gj (x)]χ(x − xj ),
j=1
also admits a uniform approximation by rational functions obtained as follows:
Given δ > 0, the increasing sequence of functions
1
ψn (x) := 1 −
1 + (1 + x)2n
is decreasing and tends to 0 on [−1, 0), and it is increasing and tends to 1 on (0, 1].
By Dini’s theorem, it is uniformly convergent to the function χ on {δ ≤ |x| ≤ 1}.
Every linear function gj+1 − gj vanishes at xj and it is easily seen that
[gj+1 (x) − gj (x)]ψn (x − xj ) → [gj+1 (x) − gj (x)]χ(x − xj )
uniformly on [0, 1] as n → ∞. Hence
Rn (x) := g1 (x) +
m−1
X
[gj+1 (x) − gj (x)]ψn (x − xj )
j=1
are rational functions such that Rn → f uniformly on [0, 1].
As said in a footnote of the Mittag-Leffler 1900 paper [ML], Phragnén observed
in 1886, being 23 years old, that curiously Runge did not see that as an application
of his theorem he could easily obtain the approximation by polynomials.
8. 1902: Fejér and approximation by averages of Fourier sums
Fejér 11 gave a new proof of THEOREM 1, in two steps as Weierstrass: (A) The
continuous function f : [0, 1] → R, supposed to be such that f (0) = f (1) = 0
and then periodized, can be approximated by the averages of its Fourier sums, and
(B) The Fourier sums, which obviously are entire functions, are then approximated
using Taylor polynomials.
11Lipót
Fejér (1880 Pécs -1959 Budapest), then called Leopold Weiss, had some difficulties in
Secondary schooling and his father took him out of school and temporally became his teacher. His
attitude in front of mathematics changed completely when he met a new teacher.
After entering in 1897 the Polytechnic University of Budapest, he spent the year 1899-1900 in
Berlin, where, after his discussions with Hermann Schwarz and being 17 years old, he proved the
“Fejér’s theorem”, published in the paper [F1] “Sur les fonctions bornées et intégrables’.
In Berlı́n, after changing his Jewish name to Lipót Fejér to claim his Hungarian culture, Schwarz
refused to talk to him.
His doctoral thesis, presented to the University of Budapest in 1902 based on his theorem,
contained his proof of THEOREM 1 and other important applications that were also included in
his paper [F2], as the facts that if the Fourier series of a function converges in a continuity point
of the function, then it converges to the value of the function in that point, and that the solution
for Dirichlet’s problem for the circle is given by the Poisson.
With this results, Fejér restored the role of Fourier series in analysis, lost after the pathological
examples in real analysis produced at that time.
Despite the reputation that Fejér had won, he remained relatively unknown in Hungary. In
1905 Henri Poincaré went to Budapest to receive the first prize Bolyai. In his welcome to the
station where he was asked Fejér. The response being ”Who is Fejr?”, Poincaré replied that he
was the greatest Hungarian mathematician and one of the greatest mathematicians in the world.
After that, Fejér was appointed professor in Kolozsvár (today Cluj, Romania) and in 1911 he was
appointed to the chair of mathematics at the University of Budapest.
20
JOAN CERDÀ
The basic step (A) is Fejér’s theorem that, for our function f , states that the
averages
N −1
1 X
σN (f, x) =
sn (f, x)
N n=0
of the Fourier sums,
N
A0 X
sN (f, x) =
+
An cos(2πnx) + Bn sin(2πnx) ,
2
n=1
uniformly approximate f .
As we have recalled,
Z 1/2
f (t)DN (x − t) dt
sN (f, x) =
−1/2
with DN as in (8),
DN (t) = 1 + 2
N
X
cos(2πnt),
n=1
Dirichlet had proved in 1829 that sN (f, x) → f (x) if f is piecewise monotone. But, as we will see below, in 1873 Paul Paul
Fejér
Bois-Raymond presented his counterexample of a continuous
function lacking this property.
It was well known that taking averages smooths fluctuations of sequences and
Fejér observed that the averages of Fourier sums can be written as
Z 1/2
Z 1/2
−1
h1 N
i
X
f (t)
σN (f, x) =
f (t)FN (x − t) dt,
Dn (x − t) dt =
N n=0
−1/2
−1/2
where the integral kernels FN are trigonometric polynomials such that
(1) FN ≥ 0,
(2) FN (−t) = FN (t),
R 1/2
(3) −1/2 FN (t) dt = 1, i
(4) limN →∞ maxδ≤t≤1/2 FN (t) = 0 si 0 < δ < 1/2,
typical properties of approximations of the Ridentity that, as in the case of Weier1
strass kernels, give the uniform convergence 0 f (t)FN (x − t) dt → f (x) si N → ∞,
R1
and the functions 0 f (t)FN (x − t) dt are also trigonometric polynomials.
This proof, similar to the on given by Lerch, is included for instance in [Ap].
As previously mentioned, in 1916 Fejér gave in [F3] another proof of THEOREM 1
by interpolation.
9. 1908: Landau presents a simple proof
In 1908, Edmund Landau, mathematical grandson of Weierstrass (Frobenius had
been his advisor),12 presented in [La] the simplest and most elementary proof of
12Landau
(1877 - 1938) was a Jew who worked in complex analysis and in number theory (he
wrote the first systematic treatment of analytic number theory, “Handbuch der Lehre von der
Verteilung der Primzahlen”). He taught at the University of Berlin from 1899 until 1909. He was
professor at Gttingen from 1909, and he also taught at the Hebrew University of Jerusalem.
WEIERSTRASS AND UNIFORM APPROXIMATION
21
THEOREM 1. It is a direct proof and in one single step that has been included in
several textbooks, such as Rudin’s [Ru] “Principles on Mathematical Analysis”.
For the proof we can suppose that f (0) = f (1) = 0, that f is extended by zero to
the whole line, and that |f | ≤ 1.
On [−1, 1] consider the even polynomials, ≥ 0,
Qn (x) = cn (1 − x2 )n ,
R1
R1
with cn so that −1 Qn = 1. Hence, cn = 1/ −1 (1 − x2 )n dx.
√
Then cn < n follows from Bernoulli’s inequality (1 + h)n ≥
1 + nh with h ≥ −1, obtained by observing that, if h > −1,
the derivative of F (h) = (1 + h)n − 1 − nh is ≥ 0, and that
F (0) = 0. In particular, (1 − x2 )n ≥ 1 − nx2 on (0, 1), and 1/cn
is estimated by
Landau
Z
1
2 n
Z
(1 − x ) dx ≥ 2
−1
√
1/ n
√
(1 − nx2 ) dx > 1/ n.
0
Every Qn is extended by zero to a function on R, and then
(a) Q
n ≥ 0,
R1
R∞
(b) −∞ Qn (x) dx = −1 Qn (x) dx = 1,
√
√
(c) If 0 < δ ≤ |x| ≤ 1, then Qn (x) < n(1 − x2 )n ≤ n(1 − δ 2 )n , so that
Qn (x) → 0 as n → ∞, uniformly if |x| ≥ δ, for every δ > 0.
Landau kernel for n = 1, 8, 16, 32
We are again in the setting of approximations of the identity. Moreover, on [0, 1],
the functions
Z ∞
Z 1
Pn (x) =
f (x − t)Qn (t) dt =
f (x − t)Qn (t) dt
−∞
∞
−1
1
Z
Z
f (t)Qn (x − t) dt =
=
−∞
f (t)Qn (x − t) dt,
0
are polynomials, since x − t ∈ [−1, 1] if x ∈ [0, 1] when 0 ≤ t ≤ 1.
22
JOAN CERDÀ
Now the proof follows as usual using the uniform continuity of f , so that |f (x) −
f (y)| < ε if |x − y| ≤ δ. For every 0 ≤ x ≤ 1,
Z
1
1
Z
Pn (x) − f (x) =
f (x − t)Qn (t) dt −
−1
f (x)Qn (t) dt
−1
R δ R 1 R −δ R1
By splitting −1 in −δ + δ + −1 , and using that |f | ≤ 1 and the properties
(a)-(c) of Qn , we conclude that
Z
1
|f (x − t) − f (x)|Qn (t) dt
|Pn (x) − f (x)| ≤
−1
Z
δ
Z
≤ ε
Qn (t) dt + 4
√
≤ ε + 4 n(1 − δ 2 )n ,
−δ
1
Qn (t) dt
δ
that becomes ≤ 2ε if n is large. Hence sup0≤x≤1 |Pn (x) − f (x)| ≤ 2ε (n ≥ N ).
Also in 1908, Charles de la Vallée Poussin (who, simultaneously with Hadamard,
in 1896 had proved the prime number theorem) obtained THEOREM 2 about the
approximation of 2π-periodic function f by trigonometric polynomials using the
periodic analogues of the Landau integrals
Z
+π
cn
−π
f (t) cos2n
x − t
2
dt.
10. 1911: The probabilistic method of Bernstein
WEIERSTRASS AND UNIFORM APPROXIMATION
23
Using probabilistic methods, Bernstein13 found a very interesting proof of THEOREM 1 contained in [Be] and that essentially is as follows:
Let x ∈ [0, 1] and {Xn } a sequence of independent Bernoulli random variables
with parameter x, described by a coin with heads with probability x and tails with
probability 1 − x. Then Sn = X1 + · · · + Xn has a binomial distribution
P {Sn = k} = Cnk xk (1 − x)n−k
(k = 0, 1, . . . , n),
since there are Cnk ways to obtain k heads and n − k tails in n proves independent
trials.
P
The mean value or mathematical expectation of Sn = nk=0 kχ{Sn =k} is
Z
n
n
X
X
E(Sn ) = Sn dP =
kP {Sn = k} =
kCnk xk (1 − x)n−k ,
k=0
k=0
and the weak law of large numbers says that Sn /n → x in probability: For every
δ > 0,
n S
o x(1 − x)
n
P − x ≥ δ ≤
.
n
δ2n
Consider now the continuous function f : [0, 1] → R such that |f | ≤ 1. The
average of the composition
f (Sn /n) =
n
X
k=0
f (Sn (k)/n)χ{Sn =k} =
n
X
f (k/n)χ{Sn =k}
k=0
is
n
n
k
k
S X
X
n
Bn (f, x) := E f
=
f
P {Sn = k} =
f
Cnk xk (1 − x)n−k .
n
n
n
k=0
k=0
These are the Bernstein polynomials associated to f , and he proved that Bn f → f
as follows:
First
Z
|f (x) − Bn (f, x)| = |E(f (x)) − E(f (Sn /n))| ≤ |f (x) − f (Sn /n)| dP = I + J
13The
Ukranian mathematician Sergei Bernstein (Odessa 1880 - Moscow 1968), in his doctoral
thesis submitted in 1904 to the Sorbonne, solved Hilbert’s nineteenth problem concerning the
analyticity of the minimizers of an energy functional, showing that aC 3 solution of a nonlinear
elliptic analytic equation in 2 variables is analytic (a problem that was finally completely solved
by Ennio De Giorgi (1956, 1957), and John Nash (1957, 1958)).
Bernstein also worked on constructive function theory and mathematical foundations of genetics,
and had a very important work in Probability theory.
In 1908, de La Vallée Poussin posed the problem of the possible approximation of a polygonal
function by means of a polynomial of degree n with error less than 1/n, and the Belgium Academy
of Science offered a prize for its solution.
In 1911 Bernstein gave a very original solution using probabilistic methods l’any, introducing
what are now called the Bernstein polynomials and giving a constructive proof of THEOREM 1.
On this topic, in 1913 he presented his second doctoral thesis “About the Best Approximation
of Continuous Functions by Polynomials of Given Degree” in Russia, where a foreign doctorate was
not accepted for academic positions. Concerning this question Bernstein said that “The example of
the problem of the best approximation of the function |x|, posed by de la Valle-Poussin, reaffirms,
once again, the fact that a well-posed specific question leads to theories of a much more general
significance.”
24
JOAN CERDÀ
with
Z
|f (x) − f (Sn /n)| dP ≤ 2ε
I=
{|Sn /n−x|≤δ}
if δ > 0 is such that |f (x) − f (y)| ≤ ε when |x − y| ≤ δ (curiously he does not
explicitly refer to the uniform continuity of f and fixes a value x0 for x). By the
law of large numbers, also
Z
x(1 − x)
J=
|f (x) − f (Sn /n)| dP ≤ 2P ({|Sn /n − x| > δ}) ≤ 2
.
δ2n
{|Sn /n−x|>δ}
Consequently, if N is large,
sup |f (x) − Bn (f, x)| ≤ 2ε + 2
0≤x≤1
1
δ2n
≤ 4ε
for every n ≥ N .
For more details and for a proof of the weak law of large
numbers see e.g. [Sa].
Now we easily prove the uniform convergence limn Bn f → f
without any reference to probabilities (see [Ce]), but the polynomials Bn f were discovered thanks to the Bernstein probabilistic
method.
Remark. The construction of the sequence {Xn } of independent Bernoulli random variables with parameter x can be obtained as follows:
On Ω := {1, 0} we define the probability P such that P (1) =
x and P (0) = 1 − x.
Then, on
ΩN = {1, 0} × {1, 0} × {1, 0} × {1, 0} × · · · ,
Bernstein
we consider the family E of all the finite unions of sets πn−1 (j) (j = 0, 1; n =
1, 2, 3, . . .), with πn (j1 , j2 , . . .) = jn , and the additive function of sets Q : E → [0, 1]
such that Q(πn−1 (j)) = x si j = 1. This function is extended in a natural way to a
probability on the σ-algebra generated by E.
Now we only need to define Xn (j1 , j2 , . . .) = jn .
11. The Weierstrass students
His greatest influence was felt through his
students (among them Sofya Kovalevskaya),
many of whom became creative mathematicians.
Encyclopaedia Britannica
Weierstrass was not only one of the leading analysts of the nineteenth century,
but he was also considered as the very best teacher of advanced students. He had
many formal and informal students.
Mittag-Leffler explains in [ML2] that in 1873 he went to Paris to follow Hermite’s
course, who received him saying “Vous avez fait erreur, Monsieur, vous auriez dû
suivre les cours de Weierstrass à Berlin”. And as consequence Mittag-Leffler moved
to Berlı́n.
WEIERSTRASS AND UNIFORM APPROXIMATION
25
In the “Mathematics Genealogy Project”, Weierstrass appears as the advisor of
42 students (in boldface we indicate those that we mention on these pages):
L. Fuchs (1858), L. Königsberger (1860), H. Schwarz (1864), E. Lampe
(1864), W. Thomé (1865), T. Berner (1865), W. Biermann (1865), N. Bugaev
(Moscow State U., 1866), F. Müller (1867), G. Cantor (1867), K. Schwering (1869),
G. Frobenius (1870), L. Kiepert (1870), E. Netto (1870), H. Bruns (1871), W. Killing
(1872), S. Kovalevskaya (U. Göttingen,1874), L.Stickelberger (1874), K. Winterberg (Humboldt U. Berlin, 1874), F. Schottky (1875), G. Hettner (1877), A. Schoenflies (1877), H. von Mangoldt (1878), P. Hoyer (1879), F. Schur (1879), E. Wiltheiss
(Humboldt U. Berlin, 1879) A. Wernicke (Humboldt U. Berlin, 1879), A. Wendt
(1880), C. Runge (1880), F. Rudio (1880), A. Piltz (1881), J. Knoblauch (1882),
R. von Lilienthal (1882), H. Stahl (1882), T. Adrian (1882), K. Weltzien (Humboldt U. Berlin, 1882), M. Blasendorff (1883), Richard Müller (U. Leipzig, 1883),
E. Kötter (1884), Reinhold Müller (1884), M. Lerch (1885), W. Howe (1887).
As in the case of Mittag Leffler, Paul du Bois-Reymond, the
most cited mathematician in the 19th century, in spite of having Kummer as the advisor for his 1959 thesis “De aequilibrio
fluidorum”, he must also be considered a student of Weierstrass.
He was not in good terms with Schwarz and cannot be considered in the same group, but he shared with Weierstrass similar
mathematical interests and same concerns for rigor.
Du Bois-Reymond, in 1875, published the Weierstrass pathological function and in 1873 was first to give in his paper “Eine
neue Theorie der Convergenz und Divergenz von Reihen mit
positiven Gliedern” a striking example of a continuous func- Du Bois-Reymond
tion with a not everywhere convergent Fourier series, having
the form
f (t) = A(t) sin(ω(t)t)
(for certain A(t) → 0 and ω(t) → 0).
Weierstrass’ most remarkable student, Sofia Kovaleskaya (Moscu 1850 - Stockholm
1891), or Sonya,14 as she liked to be called, the young woman who in St. Petersburg
joined the Russian intelligentsia, which included Fedor Dostoievsky.
After her secondary schooling, Sonya decided to continue her mathematical studies at the university, but
Russian universities were closed to women. In order to
be allowed go abroad, in 1868 she arranged a fictitious
marriage with Waldemar Kovalevsky, a young paleontologist.
In Heidelberg she discovered that women could not
matriculate at the university, but she was allowed to attend lectures unofficially. There, and Königsberger, a
former student of Weierstrass, discovered her mathematical skills.
S. Kovalevskaya
In 1870 Sonya decided to continue her studies under
Weierstrass, but again he was not allowed to let Sonya
attend his classes. After evaluating the solutions to a list of problems proposed to
14The book [Co] contains a very complete description of Kovaleskaya’s work and the article [ML2]
a very interesting review of the relations between her and Weierstrass.
26
JOAN CERDÀ
Sonya, and attending to their demands, Weierstrass accepted to give her private
lessons twice a week.
In 1874, Kovalevskaya already had three articles: the
most important one about analytic solutions of partial
differential equations, containing the well known CauchyKovalevska theorem and the only one published at the
time, in Crelle’s Journal[K1], another one dealing on the
reduction of Abelian integrals to simpler elliptic integrals,
showing her knowledge of Weierstrass’s theory, that later,
in 1884, was included in Acta [K2], and the third one on
Saturn’s rings that does not appear in the letters that she
received from Weierstrass, published in [K3].
These papers were included in her dissertation submitted in 1874 to the Göttingen University. However, despite
the strong support from Weierstrass, Kovalevskaya did not
Weierstrass
obtain an academic position and returned to Russia.
In St. Petersburg the best job she could expect was
teaching arithmetic at a girls’ school and, after her father’s death, decided to act
as a true wife of Waldemar Kowalewsky. She abandoned her mathematical research
and for six years turned to writing as a theater reviewer and science and technology
reporter, and delivered herself to a frivolous life. Weierstrass, that was informed by
Txebisxef about Sonya behavior, wrote her a letter without receiving any answer.
But in 1878 Sonya, wanted to inform Weierstrass about the birth of her daughter,
and posed to him a technical question, starting a new intense exchange of letters.
In 1880 she attended a congress held in St. Petersburg. There Mittag-Leffler, who
had met Sonya a couple of years before and had been impressed by her personality,
met her again and explains in [ML2] how she decided to visit again Weierstrass.
In Berlin, she talked with Weierstrass about the integration of partial differential
equations of planetary orbits.
After three years of trying to get a professorship for her, Mittag-Leffler persuaded
the University of Stockholm to offer her a probationary position as a private docent.
Finally, after receiving the Prix Bordin of the French Academy of Sciences for her
“Mémoire sur un cas particulier du problème de la rotation d’un corps pesant autour
d’un point fixe” published in [K4], now as a widow (in 1883 Waldemar committed
suicide), in 1889 she obtained a permanent position in mathematics at Stockholm
in 1889.
She was always in contact with Weierstrass, and from 1871 to 1890 they exchanged
more than 160 letters. It is attributed Kovalevskaya having said “all my work has
been done precisely in the spirit of Weierstrass”.
On the other hand, Weierstrass wrote to her: “never have I found anyone who
could bring me such understanding of the highest aims of science and such joyful
accord with my intentions and basic principles as you”.
After Sonya’s death, Weierstrass burned her letters but fortunately his letters were
preserved by Mittag-Leffler, who promised not to publish them before Weierstrass’
death.
Their friendship, based on purely scientific interaction, became the subject of
insinuations concerning Sonia’s mathematical achievements that deeply hurt Weierstrass.
WEIERSTRASS AND UNIFORM APPROXIMATION
27
The correspondence to her and to Mittag-Leffler, H. Schwarz, Paul du BoisReymond, Königsberger, Riemann and Fuchs, is mostly devoted to mathematical
problems, but it has also been useful to know Weierstrass’ thoughts and life.
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[Be]
[B1]
[Bi]
[Bo]
[Ce]
[Cy]
[Co]
[CJ]
[Di]
[Fa]
[F1]
[F2]
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[Ha]
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[Kl]
[K1]
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WEIERSTRASS AND UNIFORM APPROXIMATION
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[W2] K. Weierstrass, Theorie der abel’schen Functionen, J. für Reine und Angewandte Math. 52 (1854), 285-380.
[W3] K. Weierstrass, Über das sogenannte Dirisletsche Pricip, Read in a meeting
of the Königliche Preussische Akademie der Wissenschaften zu Berlin, July
14, 1870.
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analytische Darstellbarkeit sogenannter willkürlicher Funktionen einer reellen
Veränderlichen, Sitzungsberichte der Akademie zu Berlin (1885), 633-639 und
789-805.
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona
E-mail address: [email protected]