This paper proves theoretically the good behaviour of the preconditioning method where the preconditioning operator computes the first derivative at intermediate grid points and then shifts the values to the original grid points. The corresponding preconditioned eigenvalues are real and positive and lie between 1 to \(\pi\) /2. An explicit formula for these eigenvalues and the corresponding eigenfunctions is given. In the last part of the paper, the results are extended to the case of pseudospectral discretizations of systems of linear scalar equations. Numerical experiments are presented for variable coefficient operators, which confirm the good properties of the preconditioning method.