R
1. Let {en }n be a sequence of measurable functions such that en ≥ 0, R en (x)dx = 1, supp en ⊂
(−δn , δn ) with δn → 0 as n → ∞. Prove that kf ∗ en − f k1 → 0 as n → ∞ for any f ∈ L1 (R).
P
2. Deduce from Wiener’s Tauberian theorem the following
of Hardy: if ∞
n=0 an is Cesaro
Presult
summable to λ and the sequence {nan }n is bounded, then ∞
a
=
λ
in
the
usual
sense.
n
n=0
For this take
(
X
e−x , if x ≥ 0,
an ,
K(x) =
S(x) =
0,
if x < 0,
n≤ex
and show that S is bounded and slowly oscillating as x → ∞.
1