The heat and wave equations

Now define Qj(p, q) to be the S H S*-valued function which is represented in local coordinates near q by the function uj(x) constructed in the previous paragraph. Since Q0(p,p) = 1, elementary estimates show that for any J the partial sum

tends to a ¿-function as t —► 0. Moreover the construction of the Uj shows that

for some smooth error term eJt (p,q). But for J > m -f-n j2, the function tJht(p,q) tends to zero in the Cm topology as t —► 0. Thus, for sufficiently large J, k{{p,q) is an approximate heat kernel of order m. As we already observed, this together with 7.11 establishes that

hi(p,q)'52tJOj (p,q) j

is an asymptotic expansion for the heat kernel, as required.