## Transcribed Text

1. Let f be a function on the circle. For each N ≥ 1 the discrete Fourier coefficients
of f are defined by
aN (n) = 1
N
X
N
k=1
f(e
2πik/N )e
−2πikn/N , for n ∈ Z.
We also let
a(n) = Z 1
0
f(e
2πix)e
−2πinx dx
denote the ordinary Fourier coefficients of f.
(a) Show that aN (n) = aN (n + N).
(b) Prove that if f is continuous, then aN (n) → a(n) as N → ∞.
G
4. Let e be a character on G = Z(N), the additive group of integers modulo N.
Show that there exists a unique 0 ≤ ` ≤ N − 1 so that
e(k) = e`(k) = e
2πi`k/N for all k ∈ Z(N).
Conversely, every function of this type is a character on Z(N). Deduce that
e` 7→ ` defines an isomorphism from ˆ to G.
[Hint: Show that e(1) is an Nth root of unity.]
5. Show that all characters on S
1 are given by
en(x) = e
2πinx with n ∈ Z,
and check that en 7→ n defines an isomorphism from Sc1 to Z.
[Hint: If F is continuous and F(x + y) = F(x)F(y), then F
R
is differentiable. To
see this, note that if F(0) 6= 0, then for appropriate δ, c =
δ
0
F(y) dy 6= 0, and
cF(x) = R δ+x
x
F(y) dy. Differentiate to conclude that F(x) = e
Ax for some A.

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.