MATH 321 April 2006 Examination: Real Variables II, Exams of Mathematics

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Marks

[9] 1. Define

(a)

∫ (^) b a f^ (x)^ dα(x) (b) a self–adjoint algebra of functions

(c) the Fourier series of a function

[16] 2. 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 ′(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 fn : [0, 1] → IR which converges point- wise, but not uniformly to zero.

[15] 3. Let f be a continuous function on IR. Suppose that f ′(x) exists for all x 6 = 0 and that f ′(x) → 3 as x → 0. Does it follow that f ′(0) exists? You must justify your conclusion.

[15] 4. Suppose that the function f : [a, b] → IR is differentiable and that there is a number D such that |f ′(x)| ≤ D

for all x ∈ [a, b]. Let P = {x 0 , x 1 , · · · , xn} be a partition of [a, b], T = {t 1 , · · · , tn} be a choice for P and S(P, T, f ) =

∑n i=1 f^ (ti)[xi^ −^ x−^1 ] be the corresponding Riemann sum. Prove that ∣ ∣∣ ∣S(P, T, f^ )^ −

∫ (^) b

a

f (x) dx

∣ ≤^ D‖P^ ‖(b^ −^ a)^ where^ ‖P^ ‖^ = max 1 ≤i≤n[xi^ −^ xi−^1 ]

[15] 5. Let {fn : [0, 1] → IR}n∈IN be a sequence of continuous functions that obey |fn(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) =

0

T (x, y) fn(y) dy

Prove that the sequence {gn}n∈IN has a uniformly convergent subsequence.

[15] 6. (a) Let H =

(x, y) ∈ IR^2

x ≥ 0 , y ≥ 0 , x^2 + 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

m=

∑^ N

n=

am,nx^2 my^2 n^ N ∈ ZZ, N ≥ 0 , am,n ∈ IR

such that sup (x,y)∈H

∣f (x, y) − g(x, y)

∣ (^) < ε

(b) Does the result in (a) hold if H is replaced by the disk

(x, y) ∈ IR^2

x^2 + y^2 ≤ 1

[15] 7. The Legendre polynomials Pn(x) : [− 1 , 1] → IR, n ∈ ZZ, n ≥ 0, are polynomials obeying (i) Pn is of degree n with the coefficient of xn^ strictly greater than zero and

(ii)

− 1 Pn(x)Pm(x)^ dx^ =

0 if n 6 = m 2 2 n+1 if^ n^ =^ m Let f : [− 1 , 1] → IR be continuous and set an = 2 n 2 +

− 1 f^ (x)Pn(x)^ dx. Prove that (a)

n=

2 2 n+1 |an|

2 ≤ ∫^1

− 1 f^ (x)

(^2) dx with equality if and only if ∑N n=0 anPn(x) converges to f in the mean as N → ∞. (b)

n=0 anPn(x) converges in the mean to^ f^ (x).

The End