Approximation in \(L_2\) by partial integrals of the multidimensional Fourier transform over the eigenfunctions of the Sturm-Liouville operator (Q721516)

The sharp Jackson inequality with optimal argument in the modulus of continuity in the space \(L^2(\mathbb{R}_+^d)\) with the multidimensional weight \(w(t)=\prod _{j=1}^dw_j(t_j),\;t=(t_1,\dots , t_d)\in \mathbb{R}_+^d,\) where \(w_j(t_j)\) for \(t_j\in \mathbb{R}_+\) are one-dimensional continuous weight functions that are positive and continuously differentiable for \(t_j>0,\;j=1,\dots ,d\), is established. The obtained result is applied for the problem of approximation in \(L^2(\mathbb{R}_+^d)\) by partial integrals of the multidimensional Fourier transform over the eigenfunctions of the Sturm-Liouville operator. The study of the optimality of the argument is carried out with the help of multidimensional Gauss quadrature formula over zeros of an eigenfunction of the Sturm-Liouville operator. The results of the article generalize a number of known results.