Iu > 0 > Iv. Conclude that f0 = 0 is neither a local maximum nor a local

minimum of J when we use the norm — . (Thus the Euler-Lagrange search

in Exercise 8.4.11 does not look at su¬ciently many ways of approaching f0 .)

(ii) Find an in¬nitely di¬erentiable function P such that P (0) = 0, P (t) >

0 for t = 0, P (1) = 0 and P (1) > 0 (so P has a strict local minimum at 1).

[The simplest such function that I can think of is a polynomial.]

(iii) De¬ne J : A ’ R by

1

P (1 ’ f (x)2 ) + f (x)2 dx.

J(f ) =

0

Show that

inf{J(f ) : f ∈ A} = 0

but that J(f ) > 0 for all f ∈ A. Show that J has a strict local minimum

at f0 = 0 when we use the norm — (that is to say, there exists a δ > 0

such that J(f ) > J(f0 ) whenever f ’ f0 — < δ). Show that we can ¬nd a

sequence fn ∈ A such that fn ’ 0 uniformly and J(fn ) ’ 0.

(iv) Find an in¬nitely di¬erentiable function G : R2 ’ R such that,

de¬ning K : A ’ R by

1

K(f ) = G(f (x), f (x)) dx,

0

we have

sup{K(f ) : f ∈ A} = 1, inf{K(f ) : f ∈ A} = 0

but 1 > K(f ) > 0 for all f ∈ A.

Exercise K.202. [11.4, P] (i) If E is a non-empty set and (Y, ρ) a complete

metric space, we write B(E) for the set of bounded functions f : E ’ Y .

The uniform distance d∞ on B(E) is de¬ned by d∞ (f, g) = supx∈E ρ(f, g).

Show that (B(E), d∞ ) is a complete metric space.

(ii) Generalise and prove Theorem 11.3.6 and Theorem 11.3.7.

(iii) Let E be a non-empty set and (Y, ρ) a complete metric space. If fn :

E ’ Y and f : E ’ Y are functions we say that fn converges uniformly to f

539

Please send corrections however trivial to twk@dpmms.cam.ac.uk

as n ’ ∞ if, given any > 0, we can ¬nd an n0 ( ) such that ρ(fn (x), f (x)) <

for all x ∈ E and all n ≥ n0 ( ).

Generalise and prove Theorem 11.4.3 and Theorem 11.4.4.

(iv) Suppose that we now drop the condition that (Y, ρ) is complete.

Which of the results above continue to hold and which fail? [Hint. Take E

to consist of one point.]

Exercise K.203. [11.4, P] (i) Show that the integral

∞

e’ux xγ’1 dx

φγ (u) =

0

converges for all u > 0 and all γ > 0. By di¬erentiating under the integral

sign, integrating by parts and solving the resulting di¬erential equation show

that

φγ (u) = Aγ (0)uγ

where Aγ is a constant depending on γ. You should justify each step of your

argument.

(ii) Obtain the result of (i) in the case when γ is an integer by integrating

by parts.

(iii) Show that the integral

∞

2 /2

e’ux e’x

ψ(u) = dx

0

converges for all real u. By di¬erentiating under the integral sign, integrating

by parts and solving the resulting di¬erential equation show that

2 /2

ψ(u) = Ae’u

for some constant A.

(iv) Obtain the result of (iii) by a change of variable. [However, the

method of (iii) carries over to the case when u is complex and the method

of (iv) does not.]

Exercise K.204. [11.4, P] A function f : R ’ R is in¬nitely di¬erentiable

everywhere and there exists a function g : R ’ R such that f (n) (x) ’ g(x)

uniformly on each bounded interval as n ’ ∞. By considering the equation

x

f (n+1) (t) dt = f (n) (t) ’ f (n) (0),

a

or otherwise, ¬nd a di¬erential equation for g and show that g(x) = Cex for

some C. Must it be true that f (x) = Cex ? Give reasons.

540 A COMPANION TO ANALYSIS

Exercise K.205. [11.4, P] Consider a sequence of functions gn : [a, b] ’ R

and a continuous function g : [a, b] ’ R. Suppose that gn (x) ’ g(x) as

n ’ ∞ for each x ∈ [a, b].

Show that gn ’ g uniformly on [a, b] if and only if, given any sequence

xn ∈ [a, b] with xn ’ x, we have gn (xn ) ’ g(x) as n ’ ∞.

If we replace [a, b] by (a, b) does the ˜if™ part remain true? Does the

˜only if™ part remain true? Would your answers be di¬erent if we insisted, in

addition, that the gn were continuous? Give proofs and counterexamples as

appropriate.

Exercise K.206. [11.4, P] (i) A continuous function f : [0, 1] ’ R has

f (1) = 1 and satis¬es

0 ¤ f (x) < 1 for all 0 ¤ x < 1

Show that

1’δ

(f (x))n dx ’ 0

n

0

as n ’ ∞. Deduce that, if the left derivative f (1) exists and is non-zero,

then

1

1

(f (x))n dx ’

n

f (1)

0

as n ’ ∞.

(ii) A di¬erentiable function g : [0, 1] ’ R satis¬es

0 ¤ g(x) ¤ 1 for all 0 ¤ x ¤ 1.

1

(g(x))n dx remains bounded as n ’ ∞, then g(x) < 1 for

Show that, if n 0

all 0 < x < 1.

Exercise K.207. (Weierstrass approximation theorem.) [11.4, T] (i)

If 1 > · > 0, de¬ne Sn : [’1, 1] ’ R by Sn (t) = (1 + · ’ t2 )n . By considering

the behaviour of

’1

1

Tn (t) = Sn (x) dx Sn (t),

’1

or otherwise, show that, given any > 0, we can ¬nd a sequence of polyno-

mials Un such that

(a) Un (t) ≥ 0 for all t ∈ [’1, 1] and all n ≥ 1,

541

Please send corrections however trivial to twk@dpmms.cam.ac.uk

(b) Un (t) ’ 0 uniformly on [’1, ’ ] ∪ [ , 1] as n ’ ∞,

1

Un (t) dt = 1 for all n ≥ 1.

(c)

’1

t

(ii) Sketch the functions given by Vn (t) = Un (s) ds and Wn (t) =

’1

t

Vn (s) ds.

’1

(iii) Show that if g : [’1, 1] ’ R is given by g(t) = 0 for t ∈ [’1, 0] and

g(t) = t for t ∈ [0, 1], then there exists a sequence of real polynomials Pn

with Pn (t) ’ g(t) uniformly on [’1, 1].

(iv) By considering functions of the form cPn (at + b), show that, if ± ∈

[’1, 1] and if g± : [’1, 1] ’ R is given by g± (t) = 0 for t ∈ [’1, ±] and

f (t) = t ’ ± for t ∈ [±, 1], then there exists a sequence of real polynomials

Qn with Qn (t) ’ g± (t) uniformly on [’1, 1].

(v) If h : [’1, 1] ’ R is piecewise linear, show that there exists a sequence

of real polynomials Rn with Rn (t) ’ h(t) uniformly on [’1, 1].

(vi) If F : [’1, 1] ’ R is continuous, show that there exists a sequence of

real polynomials Sn with Sn (t) ’ F (t) uniformly on [’1, 1].

(vi) If f : [a, b] ’ R is continuous show that there exists a sequence of

real polynomials Pn with Pn (t) ’ f (t) uniformly on [a, b].

(vii) If f : [a, b] ’ C is continuous, show that there exists a sequence of

polynomials Pn with Pn (t) ’ f (t) uniformly on [a, b].

Exercise K.208. [11.4, P, S, ‘ ] (This is easy if you see how to apply

the result of Exercise K.207 appropriately.) Show that, given a continuous

function f : [0, 1] ’ R, we can ¬nd a sequence of polynomials Pn such that

Pn (x2 ) ’ f (x) uniformly on [0, 1] as n ’ ∞.

Show, however that there exists a continuous function g : [’1, 1] ’ R

such that we cannot ¬nd a sequence of polynomials Qn such that Qn (x2 ) ’

g(x) uniformly on [’1, 1] as n ’ ∞.

Exercise K.209. [11.4, P, T] This proof of Weierstrass™s approximation

theorem is due to Bernstein and requires some elementary probability theory.

The random variable Xn (t) is the total number of successes in n indepen-

dent trials in each of which the probability of success is t. Show, by using

Tchebychev™s inequality, or otherwise, that

Xn (t) 1

’t ≥δ ¤

Pr .

nδ 2

n

Suppose that f : [0, 1] ’ R is continuous. Let pn (t) = Ef (Xn (t)/n) the

expectation of f (Xn (t)/n). Show that pn ’ f uniformly. (You will need

542 A COMPANION TO ANALYSIS

to use the fact that a continuous function on a closed interval is uniformly

continuous and bounded. You should indicate explicitly where you use these

two facts.)

Show also that pn is a polynomial of degree at most n (pn is called a

Bernstein polynomial). Deduce the result of Exercise K.207.

If you know about convex functions (for example, if you have done Exer-

cise K.39) show that, if f is convex, then pn (x) ≥ f (x) for all x ∈ [0, 1].

Exercise K.210. [11.4, P] (i) (Dini™s theorem) Let I be a bounded interval

on the real line and let fn : I ’ R [n ≥ 1] and f : I ’ R be functions such

that fn (x) ’ f (x) as n ’ ∞ for each x ∈ I. Show that if all four of the

following conditions hold

(a) fn is continuous on I for each n,

(b) f is continuous on I,

(c) for each x, the sequence fn (x) is decreasing,

(d) I is closed,

then fn ’ f uniformly on I.

(ii) Exhibit counterexamples to show that the result is false if any one of

the four conditions is omitted.

(iii) Let

1

p0 (t) = 0 and pn (t) = pn’1 (t) + (t2 ’ pn’1 (t)).

2

Show, by induction, or otherwise, that 0 ¤ pn’1 (t) ¤ pn (t) ¤ |t| for all

t ∈ [’1, 1]. Use part (i) to show that pn (t) ’ |t| uniformly on [’1, 1].

(iv) Sketch the function h : R ’ R given by h(x) = |x| + |x ’ a| with

a > 0.

(v) Use the result of (iii) to give an alternative proof of Exercise K.207 (iii).

(vi) Generalise the result of (i) as far as you can. (For example, you could

take I to be a subset of Rn or a metric space.)

Exercise K.211. (Orthogonal polynomials.) [11.4, T] Let a < b and

let w : [a, b] ’ R be a strictly positive continuous function.

(i) Show that

b

f, g = f (t)g(t)w(t) dt

a

de¬nes an inner product on the space C([a, b]) of continuous functions from

[a, b] to R. We write f = f, f 1/2 as usual.

(ii) Prove there is a sequence s0 , s1 , s2 , . . . of polynomials with real

coe¬cients such that

543

Please send corrections however trivial to twk@dpmms.cam.ac.uk

(a) sn has exact degree n,

(b) each sn has leading coe¬cient 1, and

(c) sn , sm = 0 for n = m.

(iii) Show that, if P is a polynomial of degree at most n then there exists

one and only one solution to the equation

n

P= aj s j

j=0

with aj ∈ R. Deduce, in particular, that sn , Q = 0 whenever Q is a

polynomial of degree at most n ’ 1.

(iv) Use the fact that the polynomials are uniformly dense in C([a, b])

(Weierstrass™s theorem, Exercise K.207) to show that, if f ∈ C([a, b]),

n

: aj ∈ R for 0 ¤ j ¤ n

aj s j ’ f ’0

inf

j=0

as n ’ ∞.

n

(v) Show that,if f ∈ C([a, b]), then aj sj ’ f has a unique mini-

j=0

mum, attained when

f, sj

ˆ

aj = f (j) = .

sj , s j

Show that

n

ˆ

f (j)sj ’ f ’ 0

j=0

as n ’ ∞.

(vi) By considering the expression

b

sn (t)q(t)w(t) dt,

a

where

q(t) = (t ’ t1 )(t ’ t2 ) . . . (t ’ tk )

and t1 , t2 , . . . , tk are the points of (a, b) at which sn changes sign, prove that

sn must have exactly n distinct zeros, all of which lie in the open interval

(a, b).

544 A COMPANION TO ANALYSIS

Exercise K.212. (Legendre polynomials.) [11.4, T, ‘ ] Let p0 (x) = 1

and

dn

pn (x) = n (x2 ’ 1)n

dx

for n ≥ 1. By integrating by parts, or otherwise, show that

1

pn (x)pm (x) dx = δnm γn

’1

where γn is to be found explicitly. (Recall that δnm = 0 if n = m, δnn = 1.)

Let [a, b] = [’1, 1], w(x) = 1. Show that there exists a cn (to be found

explicitly) such that, if we write sn = cn pn , then the sn obey the conditions

of Exercise K.211 (ii). We call the pn Legendre polynomials.

Show that

1 1

2n+1 (n!)2

k n

t Pn (t) dt = 0, for 0 ¤ k ¤ n ’ 1 and t Pn (t) dt = .

(2n + 1)!

’1 ’1

Exercise K.213. (Gaussian quadrature.) [11.4, T, ‘ ] Suppose x1 , x2 , . . . , xn

are distinct points in [’1, 1]. Show that, if we write

x ’ xi

ej (x) = ,

xj ’ x i

i=j

then any polynomial P of degree at most n ’ 1 satis¬es

n

P (t) = P (xj )ej (t)

j=1

for all t ∈ [’1, 1]. Deduce that we can ¬nd real numbers a1 , a2 , . . . , an such

that

n

1

P (t) dt = aj P (xj )

’1 j=1

for all polynomials of degree n ’ 1 or less.

Gauss realised that this idea could be made much more e¬ective if we

take the x1 , x2 , . . . , xn to be the zeros of the Legendre polynomials (see

Exercise K.212 and part (iv) of Exercise K.211) and this we shall do from

now on. Show that if P is a polynomial of degree 2n ’ 1 or less we can write

545

Please send corrections however trivial to twk@dpmms.cam.ac.uk

P = pn Q + R where Q and R are polynomials of degree n ’ 1. By using this

result and equation show that

n

1

P (t) dt = aj P (xj )

’1 j=1

for all polynomials of degree 2n ’ 1 or less. (For another example of a choice

involving zeros of an orthogonal polynomial, see Exercise K.48.)

By considering polynomials of the form i=j (x ’ xi )2 show that aj > 0

for each j. By considering the constant polynomial 1 show that n aj = 2.

j=1

Let us write

n 1

Gn f = aj f (xj ), and If = f (t) dt

’1

j=1

whenever f ∈ C([’1, 1]). Show that, if f, g ∈ C([’1, 1]), then

|Gn (f ) ’ Gn (g)| ¤ 2 f ’ g and |I(f ) ’ I(g)| ¤ 2 f ’ g

∞, ∞.

Use the fact that the polynomials are uniformly dense in C([’1, 1]) (Weier-

strass™s theorem, Exercise K.207) to show that

1

Gn (f ) ’ f (t) dt

’1

as n ’ ∞ whenever f ∈ C([’1, 1]).

Exercise K.214. [11.4, P] Consider the orthogonal polynomials sn of Ex-

ercise K.211. By imitating the arguments of that exercise, show that, if » is

real, then sn ’ »sn’1 has at least n ’ 1 distinct real roots and has no multiple

roots. Deduce that sn ’ »sn’1 has n distinct real roots.

(i) If P and Q are real polynomials with a common real root at x show

that there exists a real » such that P ’ »Q has a multiple root at x. Deduce

that sn and sn’1 have no common root.

(ii) If P and Q are real polynomials such that P has real roots at x and

y with x < y and Q has no real root in [x, y] show that there exists a real

» such that P ’ »Q has a multiple root at some point w ∈ (x, y). Deduce

that, between any two roots of sn , there is a root of sn’1 .

Exercise K.215. [11.4, T] (i) De¬ne fn : [’1, 1] ’ R by fn (x) = (x2 +

n’2 )1/2 . Show that fn converges pointwise to a limit F , to be found, and fn

converges uniformly to a limit f , to be found. Show that f is not everywhere

di¬erentiable.

546 A COMPANION TO ANALYSIS

(ii) By considering the integral of appropriate witch™s hats, or otherwise,

¬nd di¬erentiable functions fn : [’1, 1] ’ R such that fn (x) = 0 for x ¤ 0

and fn (x) = 1 for x ≥ 1/n. Show that fn (x) ’ 0 for each x and there is an

f such that fn (x) ’ f (x) for each x as n ’ ∞ but that f is not continuous.

(iii) Why do these results not contradict Theorem 11.4.16?

Exercise K.216. [11.4, H] Suppose that u, v : R ’ R are in¬nitely di¬er-

∞

entiable, that v(0) = 0, that u(t) ≥ 0 for all t ∈ R and that 0 u(x) dx = 1.

In this question we shall be interested in the function g : R2 ’ R de¬ned by

g(x, y) = yv(y)u(xy)

(i) Just to get an idea of what is going on, sketch g in the particular case

when v(y) = y and u(x) = 0 for x < 1 and x > 2.

(ii) Show that g has partial derivatives of all orders (thus g is in¬nitely

di¬erentiable). Show also that

∞

g(x, y) dx = v(y)

0

for all y.

∞

(iii) Show that if G(y) = g(x, y) dx then G (0) = v (0), but