PomeLemma 14.27.7.31CA87ECD1DF Every solution furnished by lemma 14.27.3, extended by zero outside $(-1,1)$, is smooth on $\mathbb R$ and vanishes to infinite order at $\pm 1$.
Lemma 14.27.731CA87ECD1DF
Proof. Let $h$ be one of the solutions in lemma 14.27.3. On either endpoint neighborhood put $\xi =h/p$; since $p>0$ on $(-1,1)$, this quotient is smooth there, and the trapping bounds place $\xi $ in a fixed compact subinterval of $(-1,1)$. Define the rational functions
\[ R_ L(p,\xi )= \frac{(1-s^2)(1+(1+s)p)}{(1+(1+\xi )p)(1-\xi )}-1, \]
\[ R_ R(p,\xi )=1- \frac{(1-s^2)(1+(1-s)p)}{(1+\xi )(1+(1-\xi )p)}. \]
The explicit formulas for the coefficients near the endpoints give
\[ h'=q\, R_ L(p,\xi )\quad \hbox{on the left},\qquad h'=q\, R_ R(p,\xi )\quad \hbox{on the right}. \]
Both rational functions are smooth, with all derivatives bounded, on a neighborhood of the relevant set of pairs $(p,\xi )$ with $p\geq 0$ small and $\xi $ in the compact interval above. Writing $R_{L/R}$ for the function corresponding to the endpoint under consideration, we obtain from $h'=p'\xi +p\xi '$ and $q=p'$
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14.27.8
\begin{equation} \xi '=\frac{p'}p\bigl (R_{L/R}(p,\xi )-\xi \bigr ). \end{equation}
For the chosen bump,
\[ \frac{p'}p=-\frac{2r}{(1-r^2)^2}. \]
This rational function and each of its derivatives grow at most like a power of the inverse distance to the endpoint. Moreover, each derivative of $p$ is $p$ times a rational function with the same property.
An induction using (14.27.8) shows that every derivative $\xi ^{(j)}$ grows at most polynomially in the inverse distance to the endpoint. The case $j=0$ is the trapping bound on $\xi $. For the induction step, differentiate the right-hand side of (14.27.8), including its term $-\xi $, $j$ times. Every resulting term is a product of a derivative of $p'/p$, a bounded partial derivative of $R_{L/R}$, and derivatives of $p$ and of $\xi $ of order at most $j$. All of these factors have polynomial bounds, by the induction hypothesis for the derivatives of $\xi $ and since every derivative of $p$ is bounded. This gives the bound for $\xi ^{(j+1)}$.
Leibniz' rule for $h=p\xi $ now gives, for each $j$, constants $C_ j,N_ j$ such that
\[ |h^{(j)}(r)|\leq C_ j p(r)(1-|r|)^{-N_ j} \]
near the endpoints. The right-hand side tends to zero at the endpoints, because $p$ decays exponentially in the inverse distance to the endpoint. Thus all derivatives of $h$ tend to zero at $\pm 1$. Extend each derivative by zero outside $(-1,1)$. By the mean value theorem, the difference quotient of $h^{(j)}$ at an endpoint tends to the limit of $h^{(j+1)}$, which is zero. Hence these extensions are the successive derivatives of the extension of $h$, which is therefore smooth and vanishes to infinite order at $\pm 1$. $\square$
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