MATH 757: Homework 1 (Due February 22)
1. Prove the following properties of the convolution:
(a) If f, g ∈ L2 (T), then f ? g is continuous.
(b) If f, g ∈ L1 (T), then f ? g ∈ L1 (T) and kf ? gkL1 ≤ kf kL1 kgkL1 .
(c) If f ∈ L1 (T) and g ∈ Lp (T), for 1 ≤ p ≤ ∞, then f ? g ∈ Lp (T) and
kf ? gkLp ≤ kf kL1 kgkLp .
(Do the cases p = ∞ and p finite separately).
(d) If f ∈ C k (T) and g ∈ C l (T), k, l ≥ 1, then f ? g ∈ C k+l (T).
(e) If P is a trigonometric polynomial and f ∈ L1 (T), then P ?f is also a trigonometric
polynomial.
p
2. (a) Let (Ω, µ) be a measure space and let f, {fn }∞
n=1 ∈ L (Ω, µ), 1 ≤ p < ∞. Assume
that fn (x) → f (x) pointwise and that kfn kLp → kf kLp .
Prove that lim kfn − f kLp = 0.
n→∞
(b) We saw in class that if the translation function of f : R → C is defined by
τt f (x) = f (x − t), then lim kτt f − f kL1 (R) = 0.
t→0
p
Prove that if f ∈ L (R), then lim kτt f − f kLp (R) = 0.
t→0
{Kn }∞
n=1
is an approximate identity and f ∈ Lp (T), 1 ≤ p < ∞, then
(c) Prove that if
kKn ? f − f kLp → 0 as n → ∞.
3. Let {KN (x)} be a family of good kernels on T (i.e. an approximate identity).
d
(a) Show that for all m ∈ Z, lim K
N (m) = 1.
N →∞
(b) Conclude from (a) that lim kKN kL2 (T) = ∞.
N →∞
(c) If FN is the N -th Féjer kernel and DN is the N -th Dirichlet kernel, calculate
kFN kL2 (T)
lim
.
N →∞ kDN kL2 (T)
Hint: Use Plancherel for (b) and (c).