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Differential equations and applications course of the University of Oxford

Past exam of the differential equations and applications course of the University of Oxford

2011

The function $\phi-\phi'$ is continuous at zero, so its average over $[-1/n,1/n]$ tends to its value there. Consequently
$$
\lim_{n\to\infty}\langle T_n,\phi\rangle=\phi(0)-\phi'(0).
$$
The <Dirac delta> satisfies $\langle\delta,\phi\rangle=\phi(0)$, and its <distributional derivative> satisfies $\langle\delta',\phi\rangle=-\phi'(0)$. Hence the <distributional convergence> is
$$
\boxed{T_n\longrightarrow\delta+\delta'.}
$$

Solution

Past exam of the differential equations and applications course of the University of Oxford / 2011 / A / 1 / a / ii / Solution

Past exam of the differential equations and applications course of the University of Oxford / 2011 / A / 1 / a / ii / Solution

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Past exam of the differential equations and applications course of the University of Oxford / 2011 / A / 1 / a / ii / Solution

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