A New Proof Of The Asymptotic Limit Of The Lp Norm Of The Sinc Function
Abstract
We improve on the inequality
1
π
∫
∞
−∞
(
sin
2
t
t
2
)
p
dt≤
1
p
–
√
,0.2cmp≥1,
showing that
1
π
∫
∞
−∞
(
sin
2
t
t
2
)
p
dt≤C(p)
3/π
−
−
−
√
p
–
√
,
with
lim
p⟶∞
C(p)=1,
and indeed that {align*} \displaystyle{\lim_{p\longrightarrow \infty}\frac{1}{\pi}\int_{-\infty}^{\infty} (\frac{\sin^2 t}{t^2})^pdt/ \frac{\sqrt{3/\pi}}{\sqrt p}=1.} {align*}