Optimal Arguments in Jackson's Inequality in the Power-Weighted Space $L_2(\mathbb{R}^d)$

This paper is devoted to the determination of the optimal arguments in the exact Jackson inequality in the space $L_2$ on the Euclidean space with power weight equal to the product of the moduli of the coordinates with nonnegative powers. The optimal arguments are studied depending on the geometry of the spectrum of the approximating entire functions and the neighborhood of zero in the definition of the modulus of continuity. The optimal arguments are obtained in the case where the first skew field is a $l_p^d$-ball for $1\le p \le 2$, and the second is a parallelepiped.