Approximation in $L_2$ by Partial Integrals of the Fourier Transform over the Eigenfunctions of the Sturm–Liouville Operator

For approximations in the space $L_2(\mathbb{R}_+)$ by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator, we prove Jackson's inequality with exact constant and optimal argument in the modulus of continuity. The optimality of the argument in the modulus of continuity is established using the Gauss quadrature formula on the half-line over the zeros of the eigenfunction of the Sturm–Liouville operator.