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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.
 Fall '09
 Garcia
 Math

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