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Unformatted text preview: Homework 5 – Math 118B, Winter 2010
Due on Thursday, March 4th, 2010
1. Show that if f ≥ 0 and if f is monotonically decreasing, and if
n n cn =
k=1 f (k ) − f (x) dx,
1 then
lim cn n→∞ exists.
2. Let φn (x) be positivevalued and continuous for all x ∈ [−1, 1], with
1 lim φn (x) dx = 1. n→∞ −1 Suppose that {φn } converges to 0 uniformly on the intervals [−1, −c]
and [c, 1] for any c > 0. Let g be a continuous function on [−1, 1].
Show that
1 lim n→∞ φn (x)g (x) dx = g (0).
−1 3. Prove that there exist constants c1 and c2 > 0 such that
N √
c
1
√ − 2 N − c1 ≤ √ 2 , as N → ∞.
n
N n=1 4. Let {fn } be a uniformly bounded sequence of functions which are
Riemannintegrable on [a, b], and deﬁne
x Fn (x) = fn (t) dt,
a a ≤ x ≤ b. Prove that there exists a subsequence {Fnk } which converges uniformly
on [a, b].
5. If f is continuous on [0, 1] and if
1 xn f (x) dx = 0,
0 prove that f (x) = 0.
1 ∀n = 0, 1, 2, · · · , 2
6. Assume that f ∈ R(α) on [a, b], and prove that there are polynomials
{Pn } such that
b lim n→∞ a f − Pn 2 dα = 0. 7. Set P0 = 0, and deﬁne
Pn+1 (x) = Pn (x) + 2
x2 − Pn (x)
,
2 Prove that
lim Pn (x) = x, n→∞ uniformly on [−1, 1]. n ≥ 1. ...
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 Fall '09
 Garcia
 Math

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