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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 −1/2 |Ht(x)| 2 dx is of the order of magnitude of t −1/2 as t → 0. More precisely, prove that t 1/2 R 1/2 −1/2 |Ht(x)| 2 dx converges to a non-zero limit as t → 0. (b) Prove that R 1/2 −1/2 x 2 |Ht(x)| 2 dx = O(t 1/2 ) as t → 0. P∞ −∞ R ∞ [Hint: For (a) compare the sum e −∞ −cn2 t with the integral e −cx2 t dx where c > 0. For (b) use x 2 ≤ C(sin πx) 2 for −1/2 ≤ x ≤ 1/2, and apply the mean value theorem to e −cx2 t

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