polygonal

Polygonal Functions

N. L. CarothersDept. of Mathematics & Statistics

Bowling Green State University

Bowling Green, OH 43403

carother @ math.bgsu.edu

Introduction

To begin, a polygonal function is a piecewise linear, continuous function; that is, a con-

tinuous function f : [ a, b ] → R is a polygonal function if there are finitely many distinct

points a = x0 < x1 < · · · < xn = b, called nodes, such that f is linear on each of the

intervals [xk−1, xk ], k = 1, . . . , n. If we fix the nodes a = x0 < x1 < · · · < xn = b, then

it’s clear that any polygonal function based on these nodes is completely determined by

its values at the nodes.

Our first result is due to Lebesgue. In fact, it’s the first observation in Lebesgue’s

first published paper “Sur l’approximation des fonctions” (complete references are given at

the end of this paper), which appeared in 1898; Lebesgue was 23 at the time. Lebesgue’s

observation is that uniformly continuous functions can be uniformly approximated by

polygonal functions. In what follows, we will use ‖ · ‖ to denote the uniform, or sup-norm;

specifically, given a continuous function f on an interval [ a, b ], we write

‖f‖ = maxa≤x≤b

|f(x)|.

Theorem. Let f : [ 0, 1 ] → R be continuous, and let ε > 0. Then, there is a polygonal

function g : [ 0, 1 ]→ R such that ‖f − g‖ < ε.

Proof. Since f is uniformly continuous, we can choose a positive integer n, sufficiently

large, so that |f(x) − f(y)| < ε whenever |x − y| < 1/n. What this means is that the

polygonal function g defined by g(k/n) = f(k/n), for k = 0, . . . , n, and g linear on each

interval(k/n, (k + 1)/n

), satisfies ‖f − g‖ < ε. (Why?)

Next we modify our approximating function: Let h be another polygonal function,

also having nodes at (k/n)nk=0, but with h(k/n) rational and with |h(k/n) − g(k/n)| < ε

for each k. It follows that ‖g − h‖ < ε and, consequently, that ‖f − h‖ < 2ε.

Since the set of all polygonal functions taking only rational values at the nodes

(k/n)nk=0, for some n, is countable (Exercise 1), this shows that C[ 0, 1 ] is separable.

2 Polygonal Functions

1. For each n, let Qn be the set of all polygonal functions that have nodes at x = k/n,

k = 0, . . . , n, and that take on only rational values at these points. Check that Qn is

a countable set, and hence the union of the Qn’s is a countable dense set in C[ 0, 1 ].

The space of polygonal functions

Fix distinct points a = x0 < x1 < · · · < xn = b in [ a, b ], and let Sn denote the set of

all polygonal functions having nodes at the xk’s. It’s not hard to see that Sn is a vector

space. (Why?) In fact, it’s relatively clear that Sn must have dimension exactly n + 1

since there are n + 1 “degrees of freedom” (each element of Sn is completely determined

by its values at the xk’s). More convincing, perhaps, is the fact that we can easily display

a basis for Sn.

2. Let a = x0 < x1 < · · · < xn = b be distinct points in [ a, b ], and let Sn be the set of all

polygonal functions having nodes at the xk’s. Show that Sn is an n + 1-dimensional

subspace of C[ a, b ] spanned by the constant function ϕ0(x) = 1 and the “angles”

ϕk+1(x) = |x − xk| + (x − xk), for k = 0, . . . , n − 1. Specifically, show that each

h ∈ Sn can be uniquely written as h(x) = c0 +∑ni=1 ciϕi(x). [Hint: Since each side

of the equation is an element of Sn, it’s enough to show that the system of equations

h(x0) = c0 and h(xk) = c0 + 2∑ki=1 ci(xk − xi−1) for k = 1, . . . , n can be solved

(uniquely) for the ci’s.]

3. Each element of Sn can be written in the form∑n−1i=1 ai|x − xi| + bx + d for some

choice of scalars a1, . . . , an−1, b, d.

It is of some interest numerically to know just how well a continuous function can be

approximated by polygonal functions through an interpolation scheme. Specifically, let’s

stick with the nodes (k/n)nk=0, n = 1, 2, . . ., associated to [ 0, 1 ]. Given f ∈ C[ 0, 1 ] and a

positive integer n, let Ln(f) denote the unique polygonal function, with nodes at (k/n)nk=0,

which agrees with f at each of these nodes.

4. Show that ‖f − Ln(f)‖ ≤ 2/n in the case f(x) = x2.

5. Let f be Lipschitz; that is, suppose that f satisfies |f(x)− f(y)| ≤ C|x− y| for some

constant C and all x, y in [ 0, 1 ]. Show that ‖f − Ln(f)‖ ≤ 2C/n. [Hint: Given x

in [ k/n, (k + 1)/n), check that |f(x) − Ln(f)(x)| = |f(x) − f(k/n) + Ln(f)(k/n) −Ln(f)(x)| ≤ |f(x)− f(k/n)|+ |f((k + 1)/n)− f(k/n)|.]

6. Show that ‖f − Ln(f)‖ ≤ 2ωf (1/n), where ωf is the modulus of continuity of f . In

particular, if f ∈ C[ 0, 1 ], then Ln(f) converges uniformly to f .

7. Prove that Ln is a linear projection from C[ 0, 1 ] onto Sn. Moreover, given f ∈ C[ 0, 1 ],

show that ‖Ln(f)‖ ≤ ‖f‖.

Polygonal Functions 3

The upshot of Exercise 3 is that each polygonal function is a linear function plus a

linear combination of translates of the function |x|. If we knew that the function |x| could

be uniformly approximated by polynomials on, say, [−1, 1 ], then we could fashion a proof

of the Weierstrass theorem. Lebesgue’s original proof uses the fact that the binomial series

√1− z = 1 + c1z + c2z

2 + c3z3 + c4z

4 + · · · , where cn =(−1) · 1 · 3 · · · (2n− 3)

2 · 4 · 6 · · · (2n),

is uniformly convergent for 0 ≤ z ≤ 1; thus,

|x| =√

1− (1− x2) = 1 + c1(1− x2) + c2(1− x2)2 + c3(1− x2)3 + · · ·

is uniformly convergent for −1 ≤ x ≤ 1 (but is no longer a power series, of course).

8. Let a0 = 1 and an =1 · 3 · · · (2n− 1)

2 · 4 · · · (2n)for n ≥ 1. Show that cn = an − an−1 and,

hence, that c1 + · · · + cn = an − a0 ≥ −1. Conclude that the binomial series for√

1− z is uniformly convergent on [ 0, 1 ]. Why does it converge to√

1− z ?

9. Here’s an alternate proof (see Rudin for more details):

(a) Define (Pn) recursively by Pn+1(x) = Pn(x) + [x − Pn(x)2]/2, where P0(x) = 0.

Use induction to show that each Pn is a polynomial.

(b) Use induction to show that 0 ≤ Pn(x) ≤ Pn+1(x) ≤√x for 0 ≤ x ≤ 1.

(c) Use Dini’s theorem to conclude that Pn(x) converges uniformly to√x on [ 0, 1 ].

(d) Pn(x2) is also a polynomial, and Pn(x2) converges uniformly to |x| on [−1, 1 ].

Since the collection of polynomials is invariant under translation, standard arguments

now show that we can approximate |x − x0| uniformly by polynomials on any interval

[ a, b ]. Specifically, if f is a continuous function on [ a, b ], and if we can find a polynomial

p that uniformly approximates the function g(t) = f((b − a) t + a

)on the interval [ 0, 1 ],

say ‖g − p‖ < ε, then the function q(x) = p((x − a)/(b − a)

)is again a polynomial, and

uniformly approximates f on [ a, b ]; in fact, it’s easy to see that ‖f − q‖ < ε. This proves

Corollary. Let f : [ a, b ] → R be continuous, and let ε > 0. Then, there is a polynomial

p such that ‖f − p‖ < ε.

This familiar result is often called Weierstrass’s First Theorem. It appeared in 1885

in the first installment of a two-part paper. The main result in the second part, called

Weierstrass’s Second Theorem, states that each continuous, 2π-periodic function can be

uniformly approximated, on the whole line, by trigonometric polynomials. Lebesgue also

addressed this second theorem in his 1898 paper; he deduced Weierstrass’s Second Theorem


from the First through purely elementary techniques. What’s more, Lebesgue pointed out

that the two results are actually equivalent; in other words, the First Theorem can likewise

be deduced from the Second. Since there are many independent proofs of the two theorems,

this is of some interest.

A somewhat simplified version of Lebesgue’s original proofs can be found in the books

by de La Vallee Poussin and by Natanson. Instead of pursuing Lebesgue’s proof, we

will next discuss Dunham Jackson’s observation, taken from Jackson’s Carus Monograph

Fourier Series and Orthogonal Polynomials, that Weierstrass’s Second Theorem already

follows from Lebesgue’s first result and a few simple calculations.

Weierstrass’s Second Theorem

To begin, we denote the collection of continuous, 2π-periodic functions f : R → R by

C2π. Please note that each element of C2π is completely determined by its values on, say,

[−π, π ]. In particular, each element of C2π is uniformly continuous (on all of R), and we

may continue to use the sup-norm; in this case,

‖f‖ = max−π≤x≤π

|f(x)|.

What Jackson tells us is that if f is a polygonal function in C2π, then the Fourier

coefficients for f satisfy |ak|, |bk| ≤ C/k2. In particular, each 2π-periodic polygonal

function is the uniform limit of its Fourier series. Since the polygonal functions are clearly

dense in C2π, this observation gives a quick proof of Weierstrass’s second theorem.

10. Let f(x) = cx + d be a linear function on the interval [ a, b ] and let n be a positive

integer. Use integration by-parts to show that∫ b

a

f(x) cosnx dx =1

n

[f(b) sinnb− f(a) sinna

]+

c

n2[

cosnb− cosna].

Let f be a polygonal function with nodes at −π = x0 < x2 < · · · < xm = π and

having slope ck in the k-th interval [xk−1, xk ]. Then,

1

π

∫ π

−πf(x) cosnx dx =

1

π

m∑k=1

∫ xk

xk−1

f(x) cosnx dx

and, according to Exercise 10,∫ xk

xk−1

f(x) cosnx dx =1

n

[f(xk) sinnxk−f(xk−1) sinnxk−1

]+ckn2[

cosnxk−cosnxk−1].

Polygonal Functions 5

The first of these expressions is easy to sum:

m∑k=1

[f(xk) sinnxk − f(xk−1) sinnxk−1

]= f(π) sinπ − f(−π) sin(−π) = 0,

while the sum of the second expression is easy to estimate:∣∣∣∣∣m∑k=1

ckn2[cosnxk − cosnxk−1

]∣∣∣∣∣ ≤ 2m

n2·max{c1, . . . , cm}.

Now M = m · max{c1, . . . , cm} is a constant depending only on f (and not on n), thus

we’ve shown that

|an| =1

π

∣∣∣∣∫ π

−πf(x) cosnx dx

∣∣∣∣ ≤ 2M

πn2.

A similar calculation will show that we also have

|bn| =1

π

∣∣∣∣∫ π

−πf(x) sinnx dx

∣∣∣∣ ≤ 2M

πn2.

That is, the Fourier coefficients for a polygonal function f are of order 1/n2. An application

of the Weierstrass M -test (and other standard arguments) shows that the Fourier series

for f converges uniformly to f on [−π, π ].

11. Let f be a polygonal (but not necessarily periodic) function on R. Fix x ∈ R, and

consider the function

g(t) =f(x+ t)− f(x)

t, for 0 < | t | ≤ π, and g(0) = 5.

Show that g is bounded and has at most one (jump) discontinuity in [−π, π ]. In

particular, both g and g2 are Riemann integrable on [−π, π ].

12. Do the same for the function ϕ(t) =f(x+ t)− f(x)

2 sin t2

, for 0 < | t | ≤ π, and ϕ(0) = 5.

13. If ϕ and ϕ2 are both integrable over [−π, π ], show that

limn→∞

∫ π

−πϕ(t) sin(n+ 1

2 ) t dt = 0.

[Hint: sin(n+ 12 ) t = cos t2 sinnt+ sin t

2 cosnt. Thus, the integral can be written as a

sum of Fourier coefficients for the functions ϕ(t) cos t2 and ϕ(t) sin t2 .]

If we denote the n-th partial sum of the Fourier series for a fixed function f by

sn(x) =a02

+n∑k=1

(ak cos kx+ bk sin kx

),


then we may appeal to the familiar Dirichlet formula (and Exercise 12) to write:

sn(x)− f(x) =1

π

∫ π

−π

[f(x+ t)− f(x)

] sin(n+ 12 ) t

2 sin t2

dt =1

π

∫ π

−πϕ(t) sin(n+ 1

2 ) t dt.

If f is a 2π-periodic polygonal function on R, then, by Exercises 11–13, this last expression

tends to 0 as n → ∞; that is, the Fourier series for f converges pointwise to f . But we

already know that the Fourier series for f is uniformly convergent (by the M -test), thus

sn(x) must actually converge uniformly to f .

As Jackson then points out, since each continuous, 2π-periodic function on the line is

uniformly close to a polygonal function, each continuous, 2π-periodic function is in turn

uniformly close to a trigonometric polynomial. That is, we’ve just proved Weierstrass’s

Second Theorem.

Theorem. Let f : R→ R be continuous and 2π-periodic, and let ε > 0. Then, there is a

trigonometric polynomial T such that ‖f − T‖ < ε.

Many modern textbooks present Weierstrass’s Second Theorem (and often even the

First) as a corollary to Stone’s version of the Weierstrass theorem. It’s interesting to

note in this regard that, according to Cheney, Stone was influenced at least in part by

Lebesgue’s paper. In particular, the fact that |x| is the uniform limit of polynomials plays

a key role in Stone’s proof.

References

E. W. Cheney, Introduction to Approximation Theory, Chelsea, 1982.

E. W. Hobson, The Theory of Functions of a Real Variable, 2 vols., Cambridge, 1950.

D. Jackson, Fourier Series and Orthogonal Polynomials, MAA, 1941.

Ch.-J. de La Vallee Poussin, Lecons sur l’Approximation des Fonctions d’une Variable Reele, Gauthier-

Villars, 1919; also available in the volume L’Approximation, by S. Bernstein and Ch.-J. de La

Vallee Poussin, Chelsea, 1970.

H. Lebesgue, “Sur l’approximation des fonctions,” Bulletin des Sciences Mathematique, 22 (1898), 278–

287.

S. Lojasiewicz, The Theory of Real Functions, Wiley, 1988.

I. Natanson, Constructive Function Theory, 3 vols., Ungar, 1964–1965; but see also Theory of Functions

of a Real Variable, 2 vols., Ungar, 1955.

W. Rudin, Principles of Mathematical Analysis, 3rd. ed., McGraw-Hill, 1976.

K. Weierstrass, “Uber die analytische Darstellbarkeit sogenannter willkurlicher Functionen einer reellen

Veranderlichen,” Sitzungsberichte der Koniglich Preussischen Akademie der Wissenshcaften zu Berlin,

(1885), 633–639, 789–805. See also, Mathematische Werke, Mayer and Muller, 1895, vol. 3, 1–37. A

French translation appears as “Sur la possibilite d’une representation analytique des fonctions dites

arbitraires d’une variable reele,” Journal de Mathematiques Pures et Appliquees, 2 (1886), 105–138.

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