Fix $\mathbf t \in \mathbb{R}\_+^d$. I am looking for a r-atomic measure $\nu$ on $\mathbb{R}\_{+}^{d}$ that solves the following 'moment' conditions.


$$\forall\, \mathbf i \in \mathbb{N}^{d} \, s.t \, \lvert\mathbf i \rvert \le n \in \mathbb N, \; \mu\_{\mathbf i}^{\mathbf{t}} = \int\limits\_{\mathbb R\_+^d} \frac{\prod\limits\_{j=1}^d r\_j^{i\_j}}{\left(1+\sum\limits\_{j=1}^{d} r\_j t\_j \right)^{\lvert \mathbf i \rvert}}\nu\left(d\mathbf r\right)$$


For given values of $\mu\_{i}^{t}$.


Is there a transformation of my equations (by a push-forward change on $\nu$ for exemple) that would translate to a more 'classical' multivariate truncated moment problem ? i.e, i want an expression without the bottom part of the fraction..


Edit : My idea is currently to increase the dimension by one, by the change of variable : 


$$\mathbf p = \left(r\_1,...,r\_d,\frac{1}{\left(1+\sum\limits\_{j=1}^{d} r\_j t\_j \right)}\right)$$


Since $\mathbf t$ is fixed, this change of variable might be bijective, and it makes $\mu\_{\mathbf{i}}^{\mathbf t}$ the $\left(i\_1,...,i\_d,\lvert\mathbf i \rvert\right)$th moment of an other random variable.. Will this work ? I am not very confident about which algorithm exists to solve such a r-atomic moment problem.



Can this be translated to a truncated multivariate moment problem?
Changing the variables from $\mathbf r=(r\_j)$ to $\mathbf s=(s\_j)$, where 
$$s\_j:=g(\mathbf r)\_j:=\frac{r\_j}{1+\sum\_i t\_ir\_i},$$
we have 
$$r\_j:=\frac{s\_j}{1-\sum\_i t\_is\_i}.$$
So, the transformation $g$ is a homeomorphism of $\mathbb R\_+^d$ onto 
$$\Sigma:=\Big\{\mathbf s\in\mathbb R\_+^d\colon \sum\_i t\_is\_i<1\Big\}.$$
Therefore, the problem can indeed be rewritten as the multivariate moment problem to find a measure $\rho$ such that 
$$\mu\_{\mathbf i}^{\mathbf{t}}=\int\_\Sigma \rho(d\mathbf s)\,
\prod\_{j=1}^d s\_j^{i\_j}\quad
\forall\, \mathbf i \in \mathbb{N}^{d} \ s.t. \ \lvert\mathbf i \rvert \le n \in \mathbb N. $$
The measure $\nu$ can then be obtained by the formula $\nu(B)=\rho(g(B))$ for all Borel $B\subseteq\mathbb R\_+^d$. 


