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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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Heat Kernel and Integral Proofs
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