Mathematics 415 2000–01
Exercises A
[Due Monday April 2nd, 2001.]
Let (ck )k∈Z be a finitely nonzero sequence of complex numbers,
X
X −2πikξ
P (z) =
ck z k ,
p(ξ) = P e−2πiξ =
ck e
k∈Z
k∈Z
1. Show that
k
X
X
ck =
even
k
ck ⇐⇒
odd
X
k
1
(−1) ck = 0 ⇐⇒ p
= 0 ⇐⇒ P (−1) = 0.
2
k
2. Show that
k
X
kck =
even
X
kck ⇐⇒
odd
k
3. Show that we have both
the form
X
k
1
(−1) kck = 0 ⇐⇒ p
= 0 ⇐⇒ P 0 (−1) = 0.
2
0
k
k
k (−1) ck
P
= 0 and
P (z) = z
−k0
k
k (−1) kck
P
1+z
2
2
= 0 if and only if P (z) is of
Q(z)
for some polynomial Q(z) and some k0 ∈ Z.
√
√
P
Show that in this case k ck = 2 ⇐⇒ Q(1) = 2.
4. Find all possible sequences (ck )k∈Z with only c0 , c1 , c2 , c3 nonzero which satisfy all the
following conditions:
X
√
ck =
2
k
X
|ck |2 = 1
k
X
ck ck−2` = 0∀` 6= 0
k
and
X
(−1)k kck = 0.
k
[Hint. Show
Find a.]
k
k (−1) ck
P
= 0. Then P (z) =
1+z 2
2
√
( 2 + a(z − 1)) for some a ∈ C.