hw2 - Homework 2 – Math 118C Spring 2010 Due on Tuesday...

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Unformatted text preview: Homework 2 – Math 118C, Spring 2010 Due on Tuesday, April 13th, 2010 1. Suppose f ∈ R on [0, A] for all A < ∞, and f (x) → 1 as x → +∞. Prove that ∞ e−tx f (x) dx = 1. lim t + t→0 0 2. Suppose 0 < δ < π , f (x) = 1 if |x| ≤ δ , f (x) = 0 is δ < |x| ≤ π , and f (x + 2π ) = f (x) for all x. (a) Compute the Fourier coefficients of f . (b) Conclude that ∞ n=1 sin(nδ ) π−δ = , n 2 0 < δ < π. (c) Prove that ∞ n=1 sin2 (nδ ) π−δ = . 2δ n 2 (d) Let δ → 0 and prove that sin x x ∞ 0 2 dx = π . 2 3. Put f (x) = x in 0 ≤ x ≤ 2π , and use Parseval’s theorem to prove ∞ n=1 1 π2 =. n2 6 4. Let Dn be the Dirichlet kernel for n ∈ N. Define 1 KN (x) = N +1 Prove that KN (x) = N Dn (x). n=0 1 1 − cos(N + 1)x , N +1 1 − cos x and that 1 2 (a) KN ≥ 0. (b) 1 2π (c) KN (x) ≤ π KN (x) dx = 1. −π 1 2 , N + 1 1 − cos δ 0 < δ ≤ |x| ≤ π. If sN = sN (f ; x) is the N th partial sum of the Fourier series of f , consider the arithmetic means: σN = Prove that σN (f ; x) = s0 + s1 + · · · + s N . N +1 1 2π π f (x − t)KN (t) dt, −π and prove Fej´r’s theorem: If f is continuous, with period 2π , then e σN (f ; x) → f (x) uniformly on [−π, π ]. ...
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This note was uploaded on 12/27/2011 for the course MATH 118c taught by Professor Garcia during the Fall '09 term at UCSB.

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hw2 - Homework 2 – Math 118C Spring 2010 Due on Tuesday...

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