Best approximation and numerical methods in solution of differential equations

Let Q(x) bee polynomial of degree q interpolating x^m at the points $\text{x}{}_{\mathrm{<mtext>i</mtext>}}\text{,}\text{i}\text{= 0, 1, /3.,}\text{q}$, where $\text{x}{}_{\mathrm{<mtext>i</mtext>}}$ are zeros of the Tchebysheff polynomial of degree q + 1 on the interval [0, 1]. If q is of order √m, then Q(x) approximates x^m well enough. This result is used to obtain a good approximation to the solution of a system of linear differential equations.