Delta-distribution composed with a function from the Fourier representation A well known representation of Dirac's delta-distribution is via the Fourier transform of distributions:
\begin{equation}
\delta[f]:=f(0)=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}xk}f(k)\mathrm{d}k\mathrm{d}x.
\end{equation}
Can this be used to define the delta distribution composed with a function $\phi:\mathbb{R}\to\mathbb{R}$ via 
\begin{equation}
(\delta\circ\phi)[f]:=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}x\phi(k)}f(k)\mathrm{d}k\mathrm{d}x?
\end{equation}
Does this make sense?
Heuristically and in a more physics-style notation, I would argue that 
\begin{equation}
\int\int e^{\mathrm{i}x\phi(k)}f(k)\mathrm{d}k\mathrm{d}x "=" \int\left(\int e^{\mathrm{i}x\phi(k)}\mathrm{d}x\right)f(k)\mathrm{d}k "=" \int\delta(\phi(k))f(k)\mathrm{d}k.
\end{equation}
If it does not make sense, what can I say about the above expression for general functions $\phi$? E.g. I would expect the above integral to be positive if $f$ is point-wise positive.
 A: What you want to do is the pull-back of distributions. And there is a theorem (cf. Hörmander 1, Theorem 8.2.4) that if the set $\{(\phi(x),\eta) \colon \phi'(x) \eta = 0\}$ and $\operatorname{WF}(\delta) = \{ (0,\eta) \colon \eta \not = 0\}$ have empty intersection, then the pull-back is well-defined.
If you are only interested in the delta-Distribution, then there is also Theorem 6.1.5 saying that for any smooth function $\phi : X \to \mathbb{R}$ with $|\phi'| \not = 0$ on $\phi = 0$, one has that $\delta^* \phi = \frac{dS}{|\phi'|}$, where $dS$ is the Euclidean surface measure on $\{\phi = 0\}$.
Even though I mentioned no integrals, this has quite strong flavour of oscillatory integrals (cf. Shubin, Chapter 1) and FIOs (Hörmander 4) to it.
Literature:
L. Hörmander - The Analysis of Linear Partial Differential Operators 1-4
M. Shubin - Pseudodifferential Operators and Spectral Theory
A: If $\phi(k)$ vanishes at $k=k_n$, $n=1,2,\ldots$, and $\phi'(k_n)\neq 0$ for all $n$, then
\begin{equation}
\int\int e^{\mathrm{i}x\phi(k)}f(k)\,\mathrm{d}k\mathrm{d}x = 2
\pi  \int\delta(\phi(k))f(k)\,\mathrm{d}k=2\pi\sum_{n}\frac{f(k_n)}{|\phi'(k_n)|},
\end{equation}
so yes, the integral is positive for point-wise positive $f$.
