Homework Set 2
Math/ECE 430
due Friday, Feb. 8, 2013
The page numbers below refer to the book by Gasquet and Witomski.
Z x+a
f (t)dt is independent of
1. p. 26, # 3.1. Show that for f periodic with period a,
x
x.
2. p. 35, # 4.3. Prove the Parseval relation for trigonometric polynomials expressed as
sines and cosines.
3. p. 35, # 4.5. If f is periodic with period a, it is also periodic with period 2a. How
are the corresponding Fourier series related?
4. p. 36, # 4.6. Find the Parseval relation for the function f (t) = exp(iπzt) and use it
to prove
∞
X
π2
1
=
2
(x
−
n)2
sin πx n=−∞
(Hint: When you have worked out your expression for cn , apply the angle-addition
formula to simplify the sine function.)
5. p. 49, # 5.11(b). Show that for any positive integer k, f ∈ Cpk [0, a] implies that
|cn (f )| ≤ K/nk for some constant K that depends on f but not on n. Consequently,
f ∈ Cp∞ [0, a] implies lim|n|→∞ |nk cn (f )| = 0 for all k.
6. p. 50, # 5.13. Given Fourier series for f and g, work out the formula for the Fourier
series of the product f g. [Note: for part a) you need the Cauchy-Schwarz inequality
|(f, g)| ≤ kf k2 kgk2 . ]
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