April 2006
MATH 321
Name
Page 2 of 12 pages
Marks
[9]
1.
Define
Rb
(a) a f (x) dα(x)
(b) a self–adjoint algebra of functions
(c) the Fourier series of a function
Continued on page 3
April 2006
[16]
2.
MATH 321
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Give an example of each of the following, together with a brief explanation of your example.
If an example does not exist, explain why not.
(a) A differentiable function which is not monotonic but whose derivative obeys |f 0 (x)| ≥ 1.
(b) Two functions f, α : [0, 1] → IR with f continuous, but f ∈
/ R(α) on [0, 1].
(c) A continuous function f : (−1, 1) → IR that cannot be uniformly approximated by a
polynomial.
(d) A monotonically decreasing sequence of functions f n : [0, 1] → IR which converges pointwise, but not uniformly to zero.
Continued on page 4
April 2006
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Page 4 of 12 pages
Continued on page 5
April 2006
[15]
3.
MATH 321
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Page 5 of 12 pages
Let f be a continuous function on IR. Suppose that f 0 (x) exists for all x 6= 0 and that
f 0 (x) → 3 as x → 0. Does it follow that f 0 (0) exists? You must justify your conclusion.
Continued on page 6
April 2006
[15]
4.
MATH 321
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Page 6 of 12 pages
Suppose that the function f : [a, b] → IR is differentiable and that there is a number D such
that
|f 0 (x)| ≤ D
for all x ∈ [a, b] . Let P P
= {x0 , x1 , · · · , xn } be a partition of [a, b], T = {t1 , · · · , tn } be a choice
n
for P and S(P, T, f ) = i=1 f (ti )[xi − x−1 ] be the corresponding Riemann sum. Prove that
Z
S(P, T, f ) −
b
a
f (x) dx ≤ DkP k(b − a)
where kP k = max [xi − xi−1 ]
1≤i≤n
Continued on page 7
April 2006
MATH 321
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Continued on page 8
April 2006
[15]
5.
MATH 321
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Let {fn : [0, 1] → IR}n∈IN be a sequence of continuous functions that obey |f n (y)| ≤ 1 for all
n ∈ IN and all y ∈ [0, 1]. Let T : [0, 1] × [0, 1] → IR be continuous and define, for each n ∈ IN,
gn (x) =
Z
1
T (x, y) fn (y) dy
0
Prove that the sequence {gn }n∈IN has a uniformly convergent subsequence.
Continued on page 9
April 2006
[15]
6.
MATH 321
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(a) Let H = (x, y) ∈ IR2 x ≥ 0, y ≥ 0, x2 + y 2 ≤ 1 . Prove that for any ε > 0 and any
continuous function f : H → IR there exists a function g(x, y) of the form
g(x, y) =
N
N X
X
am,n x2m y 2n
N ∈ ZZ, N ≥ 0, am,n ∈ IR
m=0 n=0
such that
sup f (x, y) − g(x, y) < ε
(x,y)∈H
(b) Does the result in (a) hold if H is replaced by the disk
(x, y) ∈ IR2 x2 + y 2 ≤ 1 ?
Continued on page 10
April 2006
MATH 321
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Continued on page 11
April 2006
[15]
7.
MATH 321
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The Legendre polynomials Pn (x) : [−1, 1] → IR, n ∈ ZZ, n ≥ 0, are polynomials obeying
n
(i) Pn is of degree n with the
coefficient of x strictly greater than zero and
R1
0
if n 6= m
(ii) −1 Pn (x)Pm (x) dx =
2
if n = m
2n+1
R1
f (x)Pn (x) dx. Prove that
Let f : [−1, 1] → IR be continuous and set a n = 2n+1
2
−1
R
P∞
PN
1
2
2
2
(a)
n=0 2n+1 |an | ≤ −1 f (x) dx with equality if and only if
n=0 an Pn (x) converges to
f in the mean as N → ∞.
P∞
(b)
n=0 an Pn (x) converges in the mean to f (x).
Continued on page 12
April 2006
MATH 321
Name
Page 12 of 12 pages
The End
Be sure that this examination has 12 pages including this cover
The University of British Columbia
Sessional Examinations - April 2006
Mathematics 321
Real Variables II
Time: 2 12 hours
Closed book examination
Name
Signature
Student Number
Instructor’s Name
Section Number
Special Instructions:
No calculators, notes, or other aids are allowed.
Rules Governing Formal Examinations
1. Each candidate must be prepared to produce, upon request, a Library/AMS card
for identification.
2. Candidates are not permitted to ask questions of the invigilators, except in cases
of supposed errors or ambiguities in examination questions.
3. No candidate shall be permitted to enter the examination room after the expiration
of one half hour from the scheduled starting time, or to leave during the first half hour
of the examination.
1
9
2
16
3
15
4
15
5
15
6
15
7
15
Total
100
4. Candidates suspected of any of the following, or similar, dishonest practices shall
be immediately dismissed from the examination and shall be liable to disciplinary
action.
(a) Having at the place of writing any books, papers or memoranda, calculators,
computers, audio or video cassette players or other memory aid devices, other than
those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other candidates. The plea of
accident or forgetfulness shall not be received.
5. Candidates must not destroy or mutilate any examination material; must hand
in all examination papers; and must not take any examination material from the
examination room without permission of the invigilator.