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7. For fixed x, the Gaussian kernel function f(x ) = - is the solution to Fouriers heat equation f(x|t) = 3072f(x|t), XER,
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ANSNER Given thol 2 P(xlt),xeR, t>0 ----O 2 Sx 2 C7 lo= S(x-x Limitial con dition ikzCklt) let foxt a0 dk е 277 PCxt) SE JkIntegation given kt/2 - A e - kX dk e Plxlt) 271 ikx PCko) dke 271 Cklo) ikz P(x1o) Dinvense fousrieomsfaom Chlo)S - S x-x)2 Cx-x k Cx-x Cx-x -x 21T t2 C-x dk e k+ iCx- 2T1 t (x-x)2 2 tiCx-x 2t Jdk 2 217 Conside thi Cx-x) dk e dkdu tooi Cx-x/tTT 277 1 t i j a22 P e tiy of independent CK dk 2TT 2 t x-x 211 1 277 1 e 2t

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