Question
Transcribed Text
11. Show that if u(x, t) = (f ∗ Ht)(x) where Ht is the heat kernel, and f is
Riemann integrable, then
Z 1
0
|u(x, t) − f(x)|
2
dx → 0 as t → 0.
2. Let f and g be the functions defined by
f(x) = χ[−1,1](x) = ½
1 if |x| ≤ 1,
0 otherwise, and g(x) = ½
1 − |x| if |x| ≤ 1,
0 otherwise.
Although f is not continuous, the integral defining its Fourier transform still
makes sense. Show that
f
ˆ(ξ) = sin 2πξ
πξ
and gˆ(ξ) = µ
sin πξ
πξ
2
,
with the understanding that f
ˆ(0) = 2 and gˆ(0) = 1.
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