hw5 - Homework 5 – Math 118B, Winter 2010 Due on...

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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 positive-valued 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 Riemann-integrable 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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hw5 - Homework 5 – Math 118B, Winter 2010 Due on...

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