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.