Assuming $f(t)=3t+1 = s$, I got $$\langle\delta(3t+1),\phi\rangle = \langle\delta(s),\phi(\frac{s-1}{3})\rangle = \frac{-1}{3}\langle\delta(s),\phi(\frac{s-1}{3})\rangle$$ but got stuck in here. How shall I proceed from here?
2026-04-06 07:08:53.1775459333
How to compute $\langle\delta(3t+1),\phi\rangle$ in the sense of Distribution?
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You almost got it. By definition: $\langle \delta, \phi \rangle = \int \delta(t) \phi(t) \ dt = \phi(0)$. Thus: $\int \delta(s) \phi(\frac{s-1}{3}) \ ds = \phi(\frac{s-1}{3})|_0=\phi(\frac{-1}{3})$. Integration here is an abuse of notation.