## Class of functions that is derivable from some convolution

Let $D$ and $\Omega$ be domains in $\mathbb{R}^n$. Let $q(p,x) \colon \Omega \times D \to \mathbb{R}$ be an arbitrary function from some class $Q$. We call function $f(x) \colon D \to \mathbb{R}$ derivable from $q(p,x)$ if it can be represented as some integral $$f(x) = \int\limits_{G} F(p,q(p,x)) \; d\mu(p),$$ where $\mu$ is a measure (or a differential form) on $G \subseteq \Omega$.

For example, let $D = \Omega = \mathbb{R}^n$, $q(p,x) = p \cdot x$. Then any polynomial $P_N(x) \in \mathbb{R}[x_1,\ldots,x_n]$ of degree $N$ can be represented as a sum of $m$ polynomials $P_{k}(\omega_k \cdot x) \in \mathbb{R}[y]$ iff $m \geqslant \binom{n-1}{N+n-1}$ and $(\omega_k)$ are points on unit sphere in general position.

I want to know if there is some theory that allows to find functions derivable from some $q(p,x)$. I'm also interested in questions of minimality of such sets $G$ and in questions of derivability of polynomials.

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