UNIVERSITY OF DUBLIN
MA4151
TRINITY COLLEGE
FACULTY OF S CIENCE
SCHOOL OF MATHEMATICS
JS Mathematics
SS Mathematics
Trinity Term 2001
C OURSE : 415 — F OURIER ANALYSIS AND WAVELETS
SAMPLE
PAPER
Prof. R. M. Aron & Dr. R. M. Timoney
Answer 6 questions. Please use separate answer books for sections A and B.
Logarithmic tables are available if needed.
Section A
1.
2.
3.
4.
5. For sample questions on section A, see Professor Aron’s exercise sheets.
Section B
6.
2
(a) Define operators
√ Ta , Dλ (a ∈ R, λ > 0) on L (R) by (Ta f )(x) = f (x − a)2 and
(Dλ f )(x) = λf (λx). Verify that each of the operators is an isometry of L (R)
and that they satisfy Ta Dλ = Ta/λ Dλ .
For F the Fourier transform on L2 (R), show that FDλ = D1/λ F.
P1
(b) Assuming that φ ∈ L2 (R) satisfies φ = kk=k
c D2 Tk φ and that {Tk φ : k ∈ Z}
0 k
are orthonormal in L2 (R), show that
X
1 for ` = 0
ck ck−2` =
0 for ` ∈ Z, ` 6= 0.
k∈Z
(c) Show that φ(x) = xχ[0,1) (x) + (1 − x)χ[1,2) satisfies
1
1
1
φ = √ D2 T0 φ + √ D2 T1 φ + √ D2 T2 φ
2 2
2
2 2
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7.
(a) What is meant by a compactly supported L2 (R) function?
Show that a compactly supported L2 (R) function must be in L1 (R).
P1
(b) If a dilation equation φ = kk=k
c D2 Tk φ has a compactly supported continuous
0 k
solution φ ∈ L2 (R), show that it has its support in [k0 , k1 ].
8.
(a) Explain the method of constructing wavelet bases of L2 (R) based on a multiresolution analysis.
(b) If φ satisfies a two scale dilation equation
X
φ=
ck D2 Tk φ,
k
has orthonormal translates Tk φ (k ∈ Z), gives rise to a multiresolution analysis by
V0 = the closure of the span of {Tk φ : k ∈ Z}, Vn = D2n V0 (n ∈ Z) and if
φ ∈ L1 (R) ∩ L2 (R), then show that
Z ∞
φ(x) dx = 1
−∞
9.
(a) Define convergence for an infinite product
∞
Y
an .
n=1
Also define absolute convergence and explain how to prove that absolute convegence implies convergence.
√
Pk1
−2πikξ
(b) If p(ξ) =
is a trigonometric polynomial with p(0) = 2 and
k=k0 ck e
|p(ξ)|2 + |p(ξ + 1/2)|2 ≡ 2, then show that
∞
Y
p(ξ/2n )
√
φ̂(ξ) =
2
n=1
defines a function φ̂ ∈ L2 (R)
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Course 415 Exercise Set IV
R. M. Aron 8 January, 2001
18. Let X and Y be two arbitrary sets. Prove that there is an onto function f : X → Y or that
there is an onto function f : Y → X.
19. Calculate mina,b∈R
R1
0
(x2 − a − bx)2 dx.
R1
20. Calculate max 0 x2 g(x)dx, where the maximum is taken over all measurable functions
R1
R1
R1
g : [0, 1] → R such that 0 g(x)dx = 0 xg(x)dx = 0 and 0 g 2 (x)dx = 1. Is there a relation
between this question and problem 19?
21. Let f ∈ L1 (R). Show that the function t ∈ R
fˆ(t) ∈ C is continuous. (Hint: Let
tn → t0 ∈ R. Use the definition of the Fourier transform and the Lebesgue dominated convergence theorem to show that fˆ(tn ) → fˆ(t0 ).)
22. Let f : R → R be a continuous function whose support is contained in [A, B] (i.e.
f (t) = 0 for t ∈
/ [A, B]). For s ∈ R, define fs : R → R by fs (t) ≡ f (s + t). Prove that
s∈R
fs ∈ L1 (R) is continuous.
23. For an arbitrary real number δ > 0, let gδ : R → R be a continuous function with the
following
properties: gδ (t) ≥ 0 for all t ∈ R, gδ (t) = 0 for all t ∈ R such that |t| ≥ δ, and
R∞
g
(t)dt
= 1.
−∞ δ
(a). Sketch what the graph of such a function gδ should look like.
(b). Prove that for any f ∈ C(R) and any t0 ∈ R, f ∗ gδ (t0 ) → f (t0 ) as δ → 0.
24. Let f and g be in L1 (R). Prove that f ∗ g = g ∗ f.
25. Fix k ∈ N. Calculate fˆk for fk : R → R given by
1 if |t| ≤ k
fk (t) =
.
0 if |t| > k
Find f1 ∗ f2 and f\
1 ∗ f2 .
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Course 415 Exercise Set III
R. M. Aron 15 November, 2000
12. Theorem. All subsets of R are countable.
Proof. Let S ≡ {S ⊂ R : S is countable }.
1. Clearly S is partially ordered by the relation S1 S2 ⇐⇒ S1 ⊆ S2 (for S1 , S2 ∈ S).
2. Clearly S is inductive with this partial ordering. That is, if C ⊂ S is a chain of elements in
S, then this chain has an upper bound.
3. By Zorn’s Lemma, there is a maximal element M of S.
4. Clearly M = R. Indeed, if M ( R, then there is a point t0 ∈ R\M. However, since M
is countable so is M1 = M ∪ {t0 }. And, clearly M1 ∈ S is a strictly larger element than the
maximal element M.
This contradiction completes the proof. Of course, the preceding result is false. However, can
you (i) prove each of the clear steps above which is true and (ii) disprove each of these steps
which is false.
13. Show that `p ⊂ `q for 1 ≤ p ≤ q ≤ ∞. Is the same set inclusion true for Lp and Lq , where
we take X = [0, 1] and Lebesgue measure? What if we take X = R?
14.(a). Let A ⊂ L1 [0, 1] be defined as follows:
A = {f ∈ L1 [0, 1] :
Z
f (t)dm(t) = 1}.
[0,1]
Prove that A is closed and convex in L1 [0, 1]. Show that A has infinitely many elements of
minimal norm.
(b). Let A ⊂ C[0, 1] be defined as follows:
Z
Z
f (t)dm(t) −
f (t)dm(t) = 1}.
A = {f ∈ C[0, 1] :
[0, 21 ]
[ 12 ,1]
Prove that A is closed and convex in C[0, 1], but that A has no elements of minimal norm.
15. Let S ⊂ H be an arbitrary subset of Hilbert space H. Define
S ⊥ ≡ {x ∈ H : < x, s >= 0 for all s ∈ S}.
(a). Let H = R2 , and let S = {(1, 1), (1, 3)}. Calculate S ⊥ . Repeat for H = `2 and
S = {e2n : n ∈ N}.
(b). Prove that S ⊥ is a closed subspace of H.
(c). Show that S ⊂ S ⊥⊥ . Under what conditions do we have equality?
16. Show that c (of exercise 7) and c0 are isomorphic. That is, show that there is a linear,
continuous, bijective mapping T : c → c0 .
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17. Suppose that H is a Hilbert
space and that (ej ) is an orthonormal sequence in H (i.e. for
0 if j 6= k
each j and k, < ej , ek >=
). Prove that (ej ) is a basis for H if and only if the
1 if j = k
vector space span of {ej : j ∈ N} is dense in H.
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Course 415 Exercise Set II
R. M. Aron 13 October, 2000
6.(a). Calculate limm→∞ limn→∞ | cos(m!πx)|n .
(b). Why did I ask this question?
7. Let c ≡ {x = (xj ) : limj→∞ xj exists }. Show that c is a vector space, and that it is a
Banach space with the norm ||x|| = supj |xj |. Find a basis for c.
8. Consider the following mapping T : R2n → R2n .
x1 + x2 x3 + x4
x2n−1 + x2n x1 − x2
x2n−1 − x2n
√
√
T (x1 , x2 , x3 , ...) ≡ ( √ , √ , ...,
, √ , ...,
).
2
2
2
2
2
Prove that T is linear and also that T is an isometry, that is that ||T (x)|| = ||x|| for every
x = (x1 , .., x2n ) ∈ R2n .
9. Let X be a Banach space
basis {ej }. Prove that the mapping pi : X → K, pi (x) = xi
Pwith
∞
is continuous. (Here, x = j=1 xj ej .)
10. Recall that the space of continuous real-valued functions on [0, 1], C[0, 1], is complete (i.e. a
Banach space) when it is endowed with the norm ||f || = max0≤t≤1 |f (t)|. Let P [0, 1] ⊂ C[0, 1]
be the subspace consisting of the restrictions to [0, 1] of all polynomials.
(a). Find a discontinuous linear form φ : P [0, 1] → R.
(b*). Find a discontinuous linear form φ : C[0, 1] → R.
11. Let (fj )∞
j=1 and f be functions on [0, 1]. Consider the following properties:
(a). fj → f in L∞ [0, 1].
(b). fj → f in L2 [0, 1].
(c). fj → f in L1 [0, 1].
(d). fj (t) → f (t) for every t ∈ [0, 1].
(e). fj → f almost everywhere.
Find all possible relations between these 5 different types of convergence. Where an implication is true, prove it. Where it is false, provide a counterexample.
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MA4151
Course 415 Exercise Set I
R. M. Aron 13 October, 2000
1. A set S in a metric space (X, d) is called a Gδ provided S can be written as a countable
intersection of open sets; that is, S = ∩∞
n=1 Un , where each Un is open. Let (X, d) = (R, usual
metric d). In each case, explain whether the set S is a Gδ .
(a). S = Z.
(b). S = all irrationals.
(c). S = ∅.
(d). S = Q.
(e). S = any closed set.
2. Suppose that F is some set of continuous functions f : R → R, having the following property: For each t ∈ R, there is a constant Mt > 0 such that for every f ∈ F, |f (t)| ≤ Mt .
Prove that there is a constant M and an open interval I ⊂ R such that for all f ∈ F and all
t ∈ I, |f (t)| ≤ M.
3(*). Is there a sequence (Ij ) of non-empty, disjoint, closed subintervals of I = [0, 1] such that
I = ∪∞
j=1 Ij ?
4. Let X = R2 . For each p, 1 ≤ p < ∞, define a norm on X as follows: For x = (x1 , x2 ) ∈ X,
1
set ||(x1 , x2 )||p ≡ [|x1 |p + |x2 |p ] p . Let Bp be the closed unit ball in this norm, i.e. Bp = {x =
(x1 , x2 ) : ||x||p ≤ 1}.
(a). Draw B1 , B2 , and Bp where p is a very large number.
Let’s define a further norm on X by ||x||∞ ≡ max{|x1 |, |x2 |}.
(b). Draw B∞ . (If things have worked out correctly, Bp should ‘tend’ to B∞ . Can this
statement be made precise?)
(c). For each p ∈ [1, ∞], find all points x of the form x = (x1 , 1) such that ||x||p = 1.
5.(a). For f ∈ C[0, 1] and p ∈ [1, ∞), define
Z
||f ||p ≡ [
1
|f (t)|p dt]1/p ,
0
and
||f ||∞ ≡ max |f (t)|.
{0≤t≤1}
Let f (t) = t and g(t) = 3t − 1. Calculate ||f − g||1 , ||f − g||2 , and ||f − g||∞ .
(b). Provide the details to show that the space C[0, 1] of continuous real-valued
functions
R1
on [0, 1] is a normed space but not a Banach space, when normed by ||f || = 0 |f (t)|dt.
c UNIVERSITY OF DUBLIN 2001