Davidson K.R., Donsig A.P. Real Analysis with Real Applications

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CHAPTER 14

Fourier SeriesandApproximation

It is a natural problem to take a wave output and try to decompose it into its har-

monic parts. Engineers are able to do this with an oscilloscope. A real difﬁcultly

occurs when we try to put the parts back together. Mathematically, this amounts

to summing up the series obtained from decomposing the original wave. In this

chapter, we examine this delicate question: Under what conditions does a Fourier

series converge? We begin with L

approximation and convergence, which has a

very clean answer. Nice applications of this include the isoperimetric inequality

and sums of various interesting series. Then we turn to the more subtle questions

of pointwise and uniform convergence. The idea of kernel functions, analogous

to the Poisson kernel from the previous chapter, provides an elegant method for

understanding these notions of convergence.

14.1. Least Squares Approximations

Approximation in the L

norm is important because it is readily computable.

Also, the partial sums of the Fourier series are well behaved in this norm, unlike

pointwise and uniform convergence, which we study later.

For the purposes of applications, we need to consider piecewise continuous

functions on [a, b], as given in Deﬁnition 5.2.4. Recall that there is a partition a =

< x

< ··· < x

= b of [a, b] so that f is continuous on each interval(x

, x

i+1

)

and one-sidedlimits existat each node. Note that a piecewisecontinuous function is

bounded by applying the Extreme Value Theorem on each interval [x

, x

i+1

]. Since

it is continuous on each of these intervals, it is also absolutely integrable. Hence

piecewise continuous 2π-periodic functions have Fourierseries. Theproduct of two

piecewise continuous functions is also piecewise continuous and thus is integrable.

So we can compute the L

inner product of two piecewise continuous functions by

extending the deﬁnition for continuous functions given in Example 7.4.4.

Given a piecewise continuous 2π-periodic function f with Fourier series

f ∼ A

∞

n=1

cosnθ + B

sinnθ,

463

464 Fourier Series and Approximation

let us denote the partial sums by S

f(θ) = A

n=1

cosnθ + B

sinnθ.

14.1.1. LEMMA. Every piecewisecontinuous 2π-periodic function f is the limit

in the L

(−π, π) norm of a sequence of trigonometric polynomials.

PROOF. Let x

= −π < x

< ··· < x

= π be a partition of f into continuous

segments. Fix n ≥ 1. Let M = kfk

∞

, and let

δ = min

32N(Mn)

i+1

− x

: 0 ≤ i < N

Deﬁne a continuous function g

on [−π, π] as follows. Let g

(x) = f (x) for

x ∈ [x

+ δ, x

i+1

−δ], 0 ≤ i < N. Also, set g

(−π) = g(π) = 0. Finally, make g

linear and continuous on each segment J

= [−π, −π + δ], J

= [x

− δ, x

+ δ],

1 ≤ i < N , and J

= [π − δ, π]. See Figure 14.1 for an example.

= −π

= π

−1

FIGURE 14.1. Piecewise continuous f with continuous approxi-

mation g

Observe that kg

∞

≤ kfk

∞

= M and therefore |f(x) − g

(x)| ≤ 2M .

Moreover, the two functions agree except on the intervals J

for 0 ≤ i ≤ N. The

total length of these intervals is 2N δ. Therefore, we can estimate

kf − g

≤

i=0

(2M)

dx ≤ 8N M

δ ≤

By Corollary 13.6.6, the 2π-periodic continuous function g

is the uniform

limit of trig polynomials. So there is a trig polynomial t

so that kg

−t

∞

Then kg

− t

≤ kg

− t

∞

as well.

kf − t

≤ kf − g

+ kg

− t

Hence f is an L

limit of trig polynomials. ¥

14.1 Least Squares Approximations 465

The main import of the following theorem is part (2), which states that the

partial sums S

f converge to f in the L

norm. Part (1) says that S

f is the

best L

approximant among all trig polynomials of degree N . Since S

f is a

trigonometric polynomial, it is continuous (and in fact C

∞

). But our result applies

to piecewise continuous functions. Since the uniform limit of continuous functions

remains continuous, this L

convergence is a weaker notion.

14.1.2. LEAST SQUARES THEOREM.

Suppose that f is a piecewise continuous, 2π-periodic function. Then

(1) If t(θ) is a trigonometric polynomial of degree N,

kf − tk

= kf − S

+ kS

f − tk

(2) lim

N→∞

kf − S

= 0.

(3)

2π

−π

|f(θ)|

dθ = kfk

= A

∞

n=1

+ B

PROOF. The L

norm comes from an inner product, namely

hf, gi =

2π

−π

f(θ)g(θ) dθ.

In Lemma 7.4.5, we have already proved that {1,

√

2cosnθ,

√

2sinnθ : n ≥ 1}

form an orthonormal set in C[−π, π], with this same inner product. Since that result

depends only on the inner product, it also shows that the same set of functions is

orthonormal in the larger inner product space of piecewise continuous, 2π-periodic

functions. Notice that

(f) = hf, 1i1 +

n=1

hf,

√

2cosnθi

√

2cosnθ + hf,

√

2sinnθi

√

2sinnθ.

Thus, by the Projection Theorem from Section 7.5, S

(f) is the best approximant

to f in the subspace spanned by {1,

√

2cosnθ,

√

2sinnθ : 1 ≤ n ≤ N}. Further,

the inequality (7.5.4) immediately establishes (1).

By Lemma 14.1.1, f is the limit of trigonometric polynomials in the L

norm.

Thus given ε > 0, choose a trig polynomial t with kf −tk

< ε. So it follows from

(1) that for n ≥ N = degt,

kf − S

≤ kf − S

≤ kf − tk

< ε.

Thus (2) holds. Finally,

kfk

= lim

N→∞

= A

∞

n=1

+ B

466 Fourier Series and Approximation

14.1.3. EXAMPLE. Recall Example 13.3.2. It was shown that

|θ| ∼

−

∞

k=0

cos(2k + 1)θ

(2k + 1)

Compute the L

norm using our formula:

2π

−π

|θ|

dθ =

−

∞

k=0

(2k + 1)

∞

k=0

(2k + 1)

The integral is easily found to be π

/3, from which we deduce that

∞

k=0

(2k + 1)

−

Hence

∞

k=1

∞

k=0

(2k + 1)

∞

k=1

(2k)

∞

k=1

Therefore,

∞

k=1

An immediate consequence of this theorem is that the sines and cosines span all

of C[−π, π]. Since they are orthogonal by Theorem 7.4.5, it follows that they are

an orthonormal basis. The last result of this section requires additional background

on Hilbert spaces, from Section 7.5. In Section 9.6, we showed that there is a

Hilbert space of functions, L

(−π, π), which is the completion of C[−π, π] in the

norm. Thanks to the abstract notion of integration developed there, we can

deﬁne, for each L

function, a Fourier series using the integration formulae given

in Deﬁnition 7.4.6.

If you have not studied Section 9.6, just consider elements of L

(−π, π) as

limits, under the L

norm, of sequences of continuous functions on [−π, π]. We

will not need the exact sense in which these limits are bona ﬁde functions. To deﬁne

a Fourier series for the limit, use the limit of the Fourier series of the continuous

functions. The following theorem shows, among other things, that the limit of the

Fourier series exists.

14.1.4. COROLLARY. The functions {1,

√

2cosnθ,

√

2sinnθ : n ≥ 1} form

an orthonormal basis for L

(−π, π). The map sending a sequence a = (a

) in

(Z) to F a := f (θ) = a

∞

n=1

√

cosnθ +

√

−n

sinnθ is a unitary map.

That is, F maps `

(Z) one-to-one and onto L

(−π, π), and kF ak

= kak

for all

a ∈ `

(Z).

14.1 Least Squares Approximations 467

PROOF. We ﬁrst deﬁne F just on the space `

of all sequences a with only ﬁnitely

many nonzero terms. Then F a is a trigonometric polynomial, and F maps `

onto

the set of all trig polynomials. Part (3) of Theorem 14.1.2 shows that kF ak

kak

for each a ∈ `

(Z). Thus F is one-to-one because kF a−F bk = ka−bk 6= 0

when a 6= b.

Theorem 7.5.8 shows that `

(Z) is complete and thus is a Hilbert space. Every

vector a is a limit of the sequence P

a =

k=−n

of vectors in `

. In particular,

this sequence is Cauchy. Therefore, for each ε > 0, there is an integer N so that

a − P

< ε for all n, m ≥ N. Consequently, the sequence of functions

F P

a = a

k=1

√

coskθ +

√

−k

sinkθ is also Cauchy because

kF P

a − F P

= kP

a − P

< ε for all n, m ≥ N.

So this sequence converges in the L

norm to an element f in L

(−π, π) (because

our deﬁnition of L

is the set of all such limits).

This function f has a Fourier series, and, for example,

= 2hf, cos kθi = lim

n→∞

2hF P

√

F e

i =

√

2 lim

n→∞

a, e

i =

√

Hence f ∼ a

∞

k=1

√

coskθ +

√

−k

sinkθ. Moreover,

kfk

= lim

n→∞

kF P

= lim

n→∞

∞

−∞

= kak

This establishes a map F which maps `

(Z) into L

(−π, π) and preserves the

norm. If (Fa

) is Cauchy in L

(−π, π), then since ka

−a

k = kF a

−F a

it follows that (a

) is Cauchy in `

(Z). If a is its limit, then F a = lim

n→∞

F a

belongs to the range. So the image space is complete. The range has been deﬁned

as the completion of the trigonometric polynomials in the L

norm, and thus is a

subspace of L

(−π, π). Part (2) of Theorem 14.1.2 shows that the range of this map

contains every continuous function. Since the range is complete, it also contains

every L

limit of continuous functions. So the range is exactly all of L

(−π, π).

This completes the proof. ¥

14.1.5. REMARK. It is easy to deduce from Appendix 13.10 that the set of

complex exponentials {e

inθ

: n ∈ Z} also forms an orthonormal basis for the

Hilbert space L

(−π, π) of complex-valued L

functions. Indeed, because there

are no

√

2’s around, the formulae are cleaner. We deﬁne a map from the Hilbert

space `

(Z) of all square summable complex sequences c to the L

function with

complex Fourier series

∞

k=−∞

ikθ

. Again, this is a unitary map. In particular, if

468 Fourier Series and Approximation

f ∼

∞

k=−∞

ikθ

, then

kfk

2π

−π

|f(θ)|

dθ =

∞

k=−∞

The proofs are the same except that we need to use complex inner products.

Exercises for Section 14.1

A. Compute the Fourier series of f (θ) = θ

− π

θ for −π ≤ θ ≤ π. Hence evaluate the

sums

∞

n=0

(−1)

(2n + 1)

and

∞

n=1

B. Evaluate

∞

n=1

C. If f ∼ A

∞

n=1

cosnθ + B

sinnθ is a continuous 2π-periodic function, prove

that lim

n→∞

= lim

n→∞

= 0.

D. Show that {e

inθ

: n ∈ Z} forms an orthonormal basis for C[−π, π].

E. Use Exercise 13.3.I to ﬁnd an orthonormal basis for C[0, π] in the given inner product.

F. (a) Compute the Fourier series of f(θ) = e

aθ

for −π ≤ θ ≤ π and a > 0.

(b) Evaluate kfk

in two ways, and use this to show that

+ 2

∞

n=1

+ n

aπ

+ e

−aπ

aπ

− e

−aπ

coth(aπ).

G. (a) Express Parseval’s Theorem in terms of the complex Fourier coefﬁcients given in

Exercise 13.10.D.

(b) Let a ∈ R \ Z, and set f(θ) = e

iaθ

for θ ∈ [−π, π]. Evaluate kfk

in two ways to

deduce that

∞

n=−∞

(a − n)

sin

aπ

H. Recall the Chebychev polynomials T

(x) = cos(n cos

−1

x). Make a change of vari-

ables in Exercise E to show that the set {T

√

: n ≥ 1} is an orthonormal basis

for C[−1, 1] for the inner product hf, gi

−1

f(x)g(x)

√

1 − x

I. Show that the map F of Corollary 14.1.4 preserves the inner product.

14.2. The Isoperimetric Problem

In this section, we provide an interesting and nontrivial application of least

squares approximation. The isoperimetric problem asks, What is the largest area

that can be surrounded by a continuous closed curve of a given length? The answer

is the circle, but a method for demonstrating this rigorously is not at all obvious.

14.2 The Isoperimetric Problem 469

Indeed, the Greeks were aware of the isoperimetric inequality. However, little

was done in the way of a rigorous proof until the work of Steiner in 1838. Steiner

gave at least ﬁve different arguments, but each one had a ﬂaw. He could not es-

tablish the existence of a curve with the greatest area among all continuous curves

of ﬁxed perimeter. This difﬁculty was not resolved for another 50 years. In 1901,

Hurwitz published the ﬁrst strictly analytic proof. It is this proof that is essentially

given here.

For convenience, we shall ﬁx the length of the curve C to be 2π. This is the

circumference of the circle of radius 1 and area π. We shall show that the circle

is the optimal choice subject to the mild hypothesis that C is piecewise C

. The

argument to remove this differentiability requirement is left to the Exercises.

Points on the curve C may be parametrized by the arc length s as (x(s), y(s))

for 0 ≤ s ≤ 2π. This is a closed curve, and thus x(2π) = x(0) and y(2π) = y(0).

Since the differential of arc length is

ds =

(s)

+ y

(s)

1/2

ds,

we have the condition

(s)

+ y

(s)

= 1.

The area A(C) is given by Green’s Theorem (see Exercise 6.4.I) as

A(C) =

2π

x(s)y

(s) ds = 2πhx, y

At this stage, we need a simple lemma for computing the Fourier series of a

derivative.

14.2.1. LEMMA. Suppose that f ∼ A

∞

n=1

cosnθ + B

sinnθ is a piece-

wise C

, 2π-periodic function. Then f

has the Fourier series

∼

∞

n=1

cosnθ − nA

sinnθ.

PROOF. The Fourier coefﬁcients of f

are obtained by integration by parts:

−π

(t) cosnt dt =

f(t) cosnt

−π

f(t)n sinnt dt = nB

Similarly,

−π

(t) sinnt dt =

f(t) sinnt

−π

−

−π

f(t)n cosnt dt = −nA

And

2π

−π

(t) dt =

2π

f(t)

−π

2π

f(π) − f(−π)

= 0. ¥

470 Fourier Series and Approximation

Since x and y are piecewise C

, they and their derivatives have Fourier series

x(s) ∼ A

∞

n=1

cosns + B

sinns

y(s) ∼ C

∞

n=1

cosns + D

sinns

(s) ∼

∞

n=1

−nA

sinns + nB

cosns

(s) ∼

∞

n=1

−nC

sinns + nD

cosns.

Let us integrate the condition x

(s)

+ y

(s)

= 1 to get

1 =

2π

(s)

+ y

(s)

= kx

+ ky

∞

n=1

+ B

+ C

+ D

The area formula yields

A(C) = 2πhx, y

i = π

∞

n=1

n(A

− B

Therefore,

π − A(C) =

∞

n=1

+ B

+ C

+ D

) − π

∞

n=1

n(A

− B

)

∞

n=1

− n)(A

+ B

+ C

+ D

∞

n=1

n(A

− 2A

+ D

+ B

+ 2B

+ C

)

∞

n=1

− n)(A

+ B

+ C

+ D

∞

n=1

n(A

− D

)

+ n(B

+ C

)

The right-hand side of this expression is clearly a sum of squares and thus is

positive. The minimum value 0 is attained only if

= A

, C

= −B

, and A

= B

= C

= D

= 0 for n ≥ 2.

14.3 The Riemann–Lebesgue Lemma 471

Moreover, the arc length condition gives

1 =

+ B

+ C

+ D

) = A

+ B

Therefore there is a real number θ such that A

= cosθ and B

= sinθ. Thus the

optimal solutions are

x(s) = A

+ cos θ cos s + sinθ sins = A

+ cos(s − θ)

y(s) = C

− sinθ cos s + cosθ sins = C

+ sin(s − θ).

Clearly, this is the parametrization of a unit circle centred at (A

, C

Finally, we should relate this proof to the historical issues discussed at the

beginning of this section. Hurwitz’s proof, as we just saw, results in an inequality

for all piecewise smooth curves in which the circle evidently attains the minimum.

It does not assume the existence of an extremal curve, avoiding this problematic

assumption of earlier proofs.

Exercises for Section 14.2

A. (a) Show that if f is an odd 2π-periodic C

function, then kfk

≤ kf

(b) Deduce that if f is a C

function on [a, b] such that f (a) = f (b) = 0, then

|f(x)|

dx ≤

b − a

(x)|

dx.

HINT: Build an odd function g on [−π, π] by identifying [0, π] with [a, b].

B. Let f be a C

function that is 2π-periodic. Prove that kf

≤ kfk

HINT: Use the Fourier series and Cauchy–Schwarz inequality.

C. (a) Consider the problem of surrounding the maximum area with a curve C of length 1

mile that begins and ends at points on a straight fence a mile long.

HINT: Reﬂect the curve in the fence.

(b) Consider the corresponding problem with a fence that makes a right angle and so

covers two sides of a large ﬁeld.

D. Suppose that two rays make an angle α ∈ (0, π). Suppose that a curve of length 1

connects one ray to the other. Find the maximum area enclosed.

HINT: Consider the effect of the transformation in polar coordinates sending (r, θ) to

(r, πθ/α) on both area and arc length and apply Exercise C.

E. Use approximation by piecewise continuous functions to extend the solution of the

isoperimetric problem to arbitrary continuous curves.

14.3. The Riemann–Lebesgue Lemma

There are continuous functions with Fourier series that do not converge at every

point. Such an example was ﬁrst found by du Bois Reymond in 1876. Further

examples have been found by Fej

er and by Lebesgue. Much more recently in

1966, Carleson solved a long-standing problem conjectured 50 years earlier by

Lusin. He showed that the Fourier series of a continuous function (and indeed any

472 Fourier Series and Approximation

function) converges for all θ except for a set of measure zero. These examples

and results are beyond the scope of this course. However, under mild regularity

conditions, we can establish convergence of the Fourier series.

We ﬁrst show that if f is C

, then the convergence result is easy. With more

work, we will be able to handle functions that are only piecewise Lipschitz.

14.3.1. THEOREM. If f is a C

, 2π-periodic function, then its Fourier series

converges absolutely and uniformly to f.

PROOF. Let f ∼ A

∞

n=1

cosnθ + B

sinnθ. Since both f and f

are C

, by

Lemma 14.2.1 we have

∼

∞

n=1

cosnθ − nA

sinnθ

and

∼

∞

n=1

−n

cosnθ − n

sinnθ.

By Lemma 13.3.1, the Fourier coefﬁcients are bounded by 2kf

. Thus

| ≤

2kf

and |B

| ≤

2kf

for n ≥ 1.

Hence as in Exercise 13.4.D, compute

cosnθ + B

sinnθk

∞

≤

4kf

Since

∞

n=1

4kf

< ∞, the Weierstrass M-test shows that this Fourier series

converges absolutely and uniformly. Thus by Corollary 13.6.4, it follows that this

uniform limit equals f. ¥

14.3.2. EXAMPLE. Consider the function f(θ) = θ

− π

θ for −π ≤ θ ≤ π.

Notice that f (−π) = f(π) = 0, whence f is a continuous 2π-periodic function.

Moreover, f

(θ) = 3θ

− π

and again we have f

(−π) = f

(π) = 2π

. So f is

. Finally, f

(θ) = 6θ. Since f

(−π) 6= f

(π), this function is not C