Approximation of the Multidimensional Jacobi Transform in $L_2$ by Partial Integrals

In the space $L_2(\mathbb{R}^d_+)$ with hyperbolic weight, the exact Jackson inequality with the optimal argument in the modulus of continuity is proved. The optimal argument is the least value of the argument in the modulus of continuity for which the exact constant in the Jackson inequality takes the minimum value. The approximation is carried out by partial integrals of the multidimensional Jacobi transform. In the study of the optimal argument, the geometry of the domain of the partial integral and the geometry of the neighborhood of zero in the definition of the modulus of continuity are taken into account. The optimal argument is obtained for the case in which the first skew field is an $l_p$-ball for $1\leq p\leq 2$ and the second, a parallelepiped.