= Solution
In the space of <tempered distributions>, this <Fourier transform> convention gives $\widehat1=2\pi\delta$ and $\widehat\delta=1$. Therefore
$$
\hat g=c+2\pi\delta,\qquad\lambda g=\lambda+\lambda c\delta.
$$
The constant <distribution> and the <Dirac delta distribution> are linearly independent: testing away from zero first determines the constant coefficient, and then testing at zero determines the delta coefficient. Hence $c=\lambda$ and $\lambda c=2\pi$, giving
$$
\boxed{(c,\lambda)=(\sqrt{2\pi},\sqrt{2\pi})\quad\text{or}\quad(-\sqrt{2\pi},-\sqrt{2\pi})}.
$$
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