13. The fact that the kernel Ht(x) is a good kernel, hence u(x, t) → f(x) at each
point of continuity of f, is not easy to prove. However, one can prove directly that
Ht(x) is “peaked” at x = 0 as t → 0 in the following sense:
(a) Show that R 1/2
dx is of the order of magnitude of t
as t → 0.
More precisely, prove that t
dx converges to a non-zero
limit as t → 0.
(b) Prove that R 1/2
dx = O(t
) as t → 0.
−∞ R ∞ [Hint: For (a) compare the sum e −∞
t with the integral e
where c > 0. For (b) use x
2 ≤ C(sin πx)
for −1/2 ≤ x ≤ 1/2, and apply the mean
value theorem to e
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