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Let $u_t(x) = t^Ne^{itx}$ for $x\geqslant 0$ and $u_t(x)=0$ elsewhere. I want to calculate the distributional limit $\lim_{t\to\infty}u_t(x)$.

How would one approach such problem. I just started a little in functional analysis.

Do we need to analyse $$\left\langle u_t,\phi \right\rangle = \int_{\mathbb{R}}u_t(x)\phi(x)dx ?$$ for $\phi \in C_0^{\infty}(\mathbb{R})$. My understanding on the subject is little.

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Hint: write $u_N,t(x):=t^Ne^{itx}$ and integrate by parts $\langle u_{N,t},\phi\rangle$. This gives a recurrence relation. To deal with the case $N=0$ do an other integration by parts.

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